Citation

## Material Information

Title:
Angles between infinite-dimensional subspaces
Creator:
Jujunashvili, Abram
Place of Publication:
Denver, Colo.
Publisher:
Publication Date:
Language:
English
Physical Description:
viii, 170 leaves : ; 28 cm

## Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of Philosophy)
Degree Grantor:
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied Mathematics
Committee Chair:
Knyazev, Andrew
Committee Members:
Mandel, Jan
Franca, Leopoldo
Lodwick, Weldon
Manteuffel, Thomas

## Subjects

Subjects / Keywords:
Angles (Geometry) ( lcsh )
Algebraic spaces ( lcsh )
Dimensional analysis ( lcsh )
Algebraic spaces ( fast )
Angles (Geometry) ( fast )
Dimensional analysis ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

## Notes

Bibliography:
Includes bibliographical references (leaves 164-170).
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by Abram Jujunashvili.

## Record Information

Source Institution:
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
66462518 ( OCLC )
ocm66462518
Classification:
LD1193.L622 2005d J84 ( lcc )

Full Text
ANGLES BETWEEN INFINITE-DIMENSIONAL SUBSPACES
by
Abram Jujunashvili
M.S., Tbilisi State University, Tbilisi, Georgia, 1978
Ph.D., Optimization, Georgian Academy of Sciences, Tbilisi, 1984
A thesis submitted to the
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics

This thesis for the Doctor of Philosophy
degree by
Abram Jujunashvili
has been approved
by
Andrew Knyazev
Weldon Lodwick
Date

Jujunashvili, Abram (Ph.D., Applied Mathematics)
Angles Between Infinite-Dimensional Subspaces
Thesis directed by Professor Andrew Knyazev
ABSTRACT
In this dissertation, we introduce angles between infinite dimensional sub-
spaces in two different ways: The first definition is based on spectra of product
of orthogonal projectors and may result, e.g., in a set of angles that fills a
whole closed interval. The second definition is based on so-called s-numbers or
Courant-Fischer numbers of operators and results in a finite number of angles or
in a monotonically nondecreasing or nonincreasing countably infinite sequence
of angles. We call the second kind of angles the discrete angles. Such angles
appear in the literature, e.g., on functional canonical analysis in statistics.
For both definitions of angles we:
investigate the basic properties of the angles and establish the relationships
between different sets of angles;
introduce the concepts of principal vectors, principal subspaces and prin-
cipal invariant subspaces and investigate their properties;
Many of our definitions appear to be new in the infinite dimensional con-
text. Several properties, e.g., the relationships between the principal invariant
subspaces and principal spectral decompositions are novel.
m

I
We investigate the changes in the angles with the change in the subspaces
and prove:
estimates for an absolute error of cosines/sines (squared) of angles between
subspaces using the gap between the changed subspaces;
majorization results for the absolute value of the difference of the cosines/sines
(squared) for the discrete angles.
These estimates generalize known results in the finite dimensional case.
We investigate a deep connection of the concept of angles between subspaces
with the Rayleigh-Ritz method, using a classical result of extending a selfadjoint
nonnegative contraction to an orthogonal projector. We obtain the estimates of
a proximity of the Ritz values with the changed trial subspace.
We show how the angles between subspaces can be used to analyze conver-
gence of iterative methods for solving linear systems originated from domain
decomposition methods.
We propose a new application of the angles between subspaces for microarray
data analysis.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Andrew Knyazev
IV

DEDICATION
To David, Irakli and Nino.

ACKNOWLEDGMENT
This thesis would not have been possible without the generous support of
I am indebted to Professor Merico Argentati for many discussions, sugges-
I am also indebted to Dr Marina Kniazeva for her help and contributing to
Chapter 8.

CONTENTS
1. Introduction........................................................... 1
1.1 Overview and Motivation ............................................. 1
1.2 Notation............................................................. 7
2. Angles Between Subspaces.............................................. 10
2.1 Preliminaries....................................................... 10
2.2 Five Parts of the Space............................................. 14
2.3 Isometries Between Subspaces........................................ 20
2.4 Definitions and Properties of the Angles............................ 24
2.5 Angles Between the Subspaces in Generic Position.................... 34
2.6 Principal Vectors.................................................. 38
2.7 Principal Subspaces................................................. 47
2.8 Principal Invariant Subspaces....................................... 57
2.9 Principal Spectral Decompositions .................................. 64
3. Discrete Angles Between Subspaces.................................. 71
3.1 Definition and Basic Properties of s-Numbers........................ 71
3.2 Discrete Angles Between Subspaces................................... 76
3.3 Principal Vectors, Principal Subspaces and Principal Invariant Sub-
spaces Corresponding to the Discrete Angles......................... 82
4. Estimates for Angles Between Subspaces.............................. 85
4.1 Perturbation Bound for the Spectrum of an Absolute Value of a linear
Operator........................................................... 85
vii

4.2 Estimate for Proximity of The Angles.......................... 87
5. Estimates for Discrete Angles Between Subspaces................. 89
5.1 Some Properties of s-Numbers that Are Useful in Obtaining Estimates 89
5.2 Estimates of Absolute Error of Sines and Cosines.............. 91
5.3 Majorization for Discrete Angles Between Subspaces ................ 95
6. Estimates for Proximity of Ritz Values.......................... 112
6.1 The Basic Estimate........................................... 112
6.2 Estimates for Proximity of Sets of Ritz Values............... 114
6.3 Estimates for Proximity of Ritz Values in Discrete case...... 118
7. Using the Angles Between Subspaces for Analysis of the Convergence
of the Domain Decomposition Algorithms............................ 124
7.1 Domain Decomposition Algorithms and the Related Error Propaga-
tion Equations ................................................. 124
7.2 The Convergence of Conjugate Gradient Method for the Equations
Corresponding to Domain Decomposition Algorithm ............... 128
8. Application of Angles Between Subspaces in Microarray Data Analysis 137
8.1 Basics of Canonical Correlation Analysis and Microarray Data Analysis 137
8.2 Affymetrix Data Analysis Algorithms.......................... 139
8.3 Matlab Software for the Analysis of Affymetrix Data.......... 142
8.4 Using Angles Between Subspaces in Affymetrix Data Analysis . . . 148
Appendix
A. Description of the used data and materials...................... 150
B. Matlab Microarray Analysis Script ................................ 155
References............................................................ 164
viii

1. Introduction
1.1 Overview and Motivation
In this dissertation we focus on the following fields of research: 1) definition
and properties of angles between infinite dimensional subspaces; 2) estimates for
the proximity of the angles between different pairs of subspaces; 3) estimates
for proximity of Ritz values of a given selfadjoint operator, corresponding to
different trial subspaces; 4) angles between subspaces in analysis of convergence
of domain decomposition methods; 5) Matlab software implementation for us-
ing the angles between subspaces in microarray data analysis. The framework
for this research embodies spectral theory of bounded selfadjoint operators in
Hilbert space (infinite or finite dimensional), the Rayleigh-Ritz procedure and
majorization theory.
The concept of principal angles between subspaces is one of the classical
mathematical ideas with many applications [40]. In functional analysis, the sine
of the largest principal angle, called the gap, bounds the perturbation of a closed
linear operator by measuring the change in its graph. The smallest nontrivial
principal angle between two subspaces determines if the sum of the subspaces
is closed. In numerical analysis, principal angles appear naturally to estimate
the approximability of eigenspaces. In statistics, the cosines of principal angles
are called canonical correlations and have applications in information retrieval
and data visualization. Principal angles are very well studied in the literature,
however, mostly in the finite dimensional case. Mainly the largest and smallest
angles have been studied in the infinite dimensional case.
1

In Chapter 2 we define and analyze the angles between subspaces based
on the spectra of corresponding operators. We investigate the basic properties
of the angles, principal vectors, principal (invariant) subspaces and principal
spectral decompositions. We also investigate the relationships between principal
invariant subspaces given by isometries.
In Section 2.2, we consider a known decomposition [24] of a Hilbert space
into an orthogonal sum of five subspaces determined by all possible intersections
of two given subspaces and their orthogonal complements. Then, we introduce
the corresponding decompositions of the orthogonal projectors onto the given
subspaces and investigate spectral properties of relevant operators.
In Section 2.3, we study two important mappings the isometries of Sz.-Nagy
[62] and of Kato [37], Davis and Kahan [12]. We investigate polar decompositions
of products of orthoprojectors and these isometries.
In Section 2.4, we define the angles from one subspace to another and
the angles between subspaces. We investigate the properties of the (sets of)
angles, such as the relationships connecting the angles between the subspaces
and their orthogonal complements. The idea of definition of angles between
infinite dimensional subspaces using the operator spectral theory appears in
[26].
In Section 2.5, we observe that the projections of the initial subspaces onto
fifth part are in generic position. This fact allows us to conclude that the zero
and right angles may not correspond to the point spectra of the appropriate op-
erators, which automatically satisfies assumptions of several propositions, given
in Section 2.4.
2

In Section 2.6, we define the principal vectors and investigate their main
properties. The cases are investgated where the properties of principal vectors
are similar to those in the finite dimensional case. Using a pair of principal
vectors of given subspaces, the pairs of principal vectors are constructed for
different pairs of subspaces. Special attention is paid to the cases, where a zero
or a right angle corresponds to the point spectra of the appropriate operators.
In Section 2.7, we generalize the definition of a pair of principal vectors to
a pair of principal subspaces. We show that the set of angles between prin-
cipal subspaces consists of one point. We prove that the principal subspaces,
corresponding to the non-right angle, have the equal dimensions. We connect
principal subspaces for the given pair of subspaces with principal subspaces for
their orthogonal complements.
In Section 2.8, we generalize the definition of a pair of principal subspaces to
a pair of principal invariant subspaces. Given a pair of principal invariant sub-
spaces of T and Q, we construct principal invariant subspaces for other pairs of
subspaces, such as !F and JrL and Q, and Q1. We investigate the ques-
tion of uniqueness of principal invariant subspaces. We show that the isometries
given in Section 2.3 map one subspace of a given pair of principal invariant sub-
spaces onto another, and vice versa. We find a connection between orthogonal
projectors onto the principal invariant subspaces.
In Section 2.9, we define and investigate principal spectral decompositions.
We show how the isometries introduced in Section 2.3 link the spectral families
of two products of projectors.
In Chapter 3, we introduce discrete angles between subspaces and investigate
3

their properties. We give different definitions of discrete angles and show their
equivalence. We investigate discrete angles between several pairs of subspaces
and relationships between the largest and smallest discrete angles.
In Section 3.1, we describe s-numbers and their basic properties, using [22].
We show that this definition is equivalent to the Courant-Fisher max-min prin-
ciple. We study the s-numbers of relevant operators.
In Section 3.2, we give two equivalent definitions of discrete angles between
subspaces: a recursive definition and a definition based on s-numbers of the
products of orthoprojectors. Such definitions appear in the literature, e.g., on
functional canonical analysis in statistics [28]. We investigate the discrete angles
between different pairs of subspaces.
In Section 3.3, similar to the general case, we define the principal vectors,
principal (invariant) subspaces and discuss their properties. The subset of dis-
crete angles is given, for which all properties of the principal vectors and prin-
cipal subspaces from Sections 2.6 and 2.7 also hold for discrete angles. For a
pair of subspaces a necessary condition is derived such that the pair is a pair of
principal invariant subspaces.
In Chapter 4, we obtain estimates for proximity of angles. In Section 4.1,
we obtain an estimate for the distance between the spectra of absolute values
of operators. This estimate alows us to obtain an estimate for the proximity of
the sines and cosines of angles between subspaces. In Section 4.2, we estimate
a proximity of squared cosines of angles from one subspace to another.
In Chapter 5, we obtain estimates for discrete angles between subspaces.
We estimate the maximal difference between the cosines, and next we obtain
4

different results on the weak majorization of the angles.
In Section 5.1, we remind the reader of the known properties of s-numbers
that are useful in the sequel. In the next two sections we generalize some results
from [40] and [42].
In Section 5.2, we estimate an absolute error of cosines/sines of the discrete
angles between two pairs of subspaces. One subspace is fixed in both pairs, and
the second one changes. We estimate these quantities by the gap (aperture)
between the inital and changed subspace. We consider the case, where both
subspaces of the initial pair are changed and we obtain an estimate, using both
gaps.
In Section 5.3, we generalize the well-known Lidskii-Mirsky-Wielandt the-
orem for matrices to the case of the Courant-Fischer numbers from the top of
bounded selfadjoint operators. We prove a majorization result for the absolute
value of the difference of s-numbers by s-numbers of the difference. We prove
that the absolute value of the difference of the cosines/sines (squared) of the
discrete angles between subspaces is weakly majorized by the sines of the angles
between the changed subspaces. The results about majorization of sines/cosines
(squared) are generalizations of corresponding results of [40], and their proofs
are also similar.
In Chapter 6, we investigate a connection between the concept of angles
between subspaces and the Rayleigh-Ritz method, based on a result of extending
a selfadjoint nonnegative contraction to an orthogonal projector. We generalize
finite dimensional estimates for a proximity of the sets of Ritz values with the
changed trial subspace to the infinite dimensional case.
5

In Section 6.1, we generalize Theorem 1 [41] to Hilbert spaces. This result
provides an estimate of the proximity of Rayleigh quotients by the sine of angle
between the vectors.
In Section 6.2, we obtain a bound on the Hausdorff distance between the Ritz
values. This result has been presented at the 12th ILAS Conference, Regina,
In Section 6.3, we prove a majorization result of the proximity of discrete
Ritz values by the sines of angles between trial subspaces. We use the technique
of extension selfadjoint nonnegative contraction to the orthogonal projector [40].
The results of this section are direct generalizations of the corresponding results
[40] to infinite dimensional case and the proofs also are similar.
In Chapter 7, we use angles between subspaces to analyze the convergence
of iterative methods for solving linear sytems originated from the domain de-
composition method. We show that in the one dimensional case, the spectra of
the error propagation operators consist of two or three points, which leads to
the finite convergence of the conjugate gradient method. We construct the pairs
of principal vectors (functions).
In Chapter 8, we discuss the canonical correlations and Affymetrix algo-
rithms for microarray data analysis. We briefly describe the canonical corre-
lation analysis and microarray technology in Section 8.1. In Section 8.2, we
descuss the algorithms that are used in Affymetrix software. In Section 8.3, we
describe our Matlab code that performs single-array and comparison analysis
for Affymetrix data. Finally, we use of angles between subspaces to analyze the
Affymetrix data in Section 8.4.
6

1.2 Notation
H
B(H)
R
u,v
T,Q
(u, V)
N!
m
PL
Pr
Â£(D
Sp(r)
Â£C(T)
dist(Si, S2)
eP(f,G)

Hilbert space.
Banach algebra of bounded linear operators on H.
Vector space of real n-tuples.
Lower case Roman letters denote vectors.
Upper Roman letters denote operators.
Calligraphic letters denote subspaces of 7i.
Hilbert space inner product.
Norm of vector in H.
Norm of operator, acting on H.
Orthogonal complement of the subspace T.
Orthogonal projector onto the subspace T.
Spectrum of the operator T.
Point spectrum of the operator T.
Continuous spectrum of the operator T.
Hausdorff distance between the sets Si and S2.
One-directional, non-symmetric angles
from P to Q.
Angles between P and Q.
Angles that correspond to eigenvalues.
Direct sum of subspaces or operators.
7

0 Intersection of the first subspace with orthogonal complement of the second one.
gap{f, Q) The gap, aperture between the subspaces T and Q.
st(T) s-numbers of the operator T.
\T\ Absolute value of an operator.
S)(T) The domain of an operator T.
ftt(T) The null-space of an operator T.
X(T) The range of an operator T.
E( A) The spectral family of an operator.
*icn The Courant-Fischer numbers from the top of the operator T.
Aim The Courant-Fischer numbers from the bottom of the operator T.
eime) The set of smallest discrete angles between subspaces T and Q.
elm, s) The set of largest discrete angles between subspaces T and Q.
S(A) The sequence of non-increasingly ordered s-numbers of bounded operator A.
T - 8

A^(.A) The set of the discrete Ritz values
from the top of the operator A.
A^(A) The set of the discrete Ritz values
from the bottom of the operator A.
Hq(Q) The Sobolev space with zero trace.
PM The Perfect Match probe.
MM The MisMatch probe.
9

2. Angles Between Subspaces
In this chapter, we define and analyze the angles between subspaces using
the spectra of corresponding operators. We investigate the basic properties of
angles, principal vectors, principal (invariant) subspaces and principal spectral
decompositions.
2.1 Preliminaries
Let H be a (real or complex) Hilbert space, T and Q be its proper nontrivial
subspaces. A subspace is defined as a closed linear manifold. Let P? and Pg be
the orthogonal projectors onto T and Q, respectively. We denote by 13(H) the
Banach space of bounded linear operators defined on H with a norm
||T|| = sup ||Tu||.
uH
MM
For T G B(H) we define |T| = (T*T)1/2, using the positive square root. T\u
denotes the restriction of the operator T to its invariant subspace U. E(T) and
Sp(T) denote the spectrum and point spectrum of the operator T, respectively.
Ep(T) is defined as (see, e.g., [16], Definition 1, p. 902) the set of numbers A in
Â£(T), for which XI T is not one-to-one, where / denotes the identity. In other
words, Ep(T) consists of eigenvalues (of finite or infinite multiplicity). 2)(T)
denotes the domain, 91(T) denotes the null space, and 9K(T) denotes the range
of the operator T. Let us mention that we do not use a Hilbert space dimension
(see, e.g., Definition 4.15, p. 17, [10]) in this dissertation. We distinguish
only the finite and infinite dimensions. If q is finite number and T is infinite
10

dimensional, then we have min{g, dim.?7} = q and max{g, dim P} = dim,?7.
Also, if p dim.?7 = oo and q = dim Q = oo, then p < q holds.
Some useful properties of orthogonal projectors on a Hilbert space H that
we need are the following (see, e.g., [13], Lemmas 4-7):
p2 = p = p* .
The product of two projectors P? and Pg is also a projector if and only if
Pjr and Pg commute, i.e., if P?Pg PqPf- In this case
PfPg = P?ng'i
Two subspaces P and Q are orthogonal if and only if P?Pg = 0;
PjrPg = Pjr if and only if P C Q .
If we represent the space TC as a direct sum of the mutually orthogonal
subspaces H = P P^, then evidently PjrP^PgPjr = P^PgPyrPT = 0 and
the block representation of the operator P^PgPjr will be
PjrPgPf =
( PtPq\t 0^
V
o
o
7
Consequently we consider only the restrictions of the operators P^PgPj: and
PgPjrPg to the subspaces P and Q that are equal to the operators PfPglf and
PqPf lei respectively.
The following known [44] lemma introduces the spectra of operators that
play the central roles later in definitions of angles between subspaces. We provide
the most interesting parts of the proof of [44] for better exposition.
11

Lemma 2.1 (Lemma 2.4, [44]) Fr orthogonal projectors P;r and Pg we have
1. Z(PrPg) = 'E{PrPgPr) C[0,1];
S. 2(Pjr-Pg)c[-l,l];
3. If A 7^ 0, 1, then A E(/y Pg) if and only if 1 A2 G T,(P?-Pg).
Proof: 1. For any complex A, the block form of XI P?Pg with respect to to
the decomposition H = IF is
(PAM PrPgPr)Pr ~PrPg(I Pjf
XI PpPg = ,
\ o Mi-PA )
which implies that Y,(PjrPg) = T,(P^Pg\jr) U {0}. From the equation XI
PjrPgPjr = Pf(XI PjrPgPj^Pyr + A (I ~ Pjr) We Conclude that E(PyrPgP^) =
T,{P^Pg\r) U {0}, and E(P?Pg) = 'L(PjrPgPjr) follows.
2. This assertion follows from the identity (see, e.g., [2], p. 70)
\\Pr PgII = ma^llP^I!, ||PSP^||}.
3. It is enough to note that, for any complex A,
[(A 1)/ + Pjr] [XI (Pjr ~ P*)][(A + 1)7 Pg] =
[(A 1)(A/ + Pg) + PrPg][{X + 1)7 Pg] =
(A 1)(A + 1)(A7 + Pg) + (A + l)PFPg (A 1)(A + 1)PC PFPg =
A[(A2 1)7 + PpPg}.
12

For subspaces T and Q of Tt let us define the quantities cq{T, Q) and c(lF, Q)
by (see, e.g., [13])
co{F,G) = sup{|(u,v)| | M < l,v G \\v\\ < 1}, (2.1)
and
c{T,G) = sup{|(u,u)| | u g Fv\ {F^G)1-, |M| < 1,
vegnifng)1, H < i}, (2.2)
which are called in [13] the cosines of the minimal angle and angle, respec-
tively between subspaces T, g.
It follows directly from the definitions of c(jF, g) and Q) that
c(F,G) The following theorem, which we use later, gives the necessary and sufficient
conditions for closeness of the range of product of two bounded linear operators
in terms of these quantities.
Theorem 2.2 ([13], Theorem 22). Let A and B be bounded linear operators
on Tt with closed ranges 93(^4) = 93(^4) and 93(5) = 93(5). Then the following
statements are equivalent:
1. AB has closed range;
2. c0(93(5),93(A) n [91(4) n 93(5)]x) < 1;
3. c(93(5),93(A)) < 1.
13

2.2 Five Parts of the Space
In this section we consider the decomposition of a Hilbert space into an
orthogonal sum of five subspaces. Then, based on the corresponding decom-
positions of the orthogonal projectors onto the considered subspaces, spectral
properties of some relevant operators are investigated.
It is known (see, e.g., [11], [24], [65]) that, given subspaces F and Q, the
space H can be represented as an orthogonal sum of the subspaces
27100 = jrnÂ£, 9jt01 = Fngx, 97i10 = jf1 ng, = g1 (2.4)
and the rest 271, where
27t = 9710 97ti (2.5)
with
971o F (971oo 971oi),
rnix = ^x(mt10fm11), (2.6)
so that, we can write
H = 97toofWoi9711o97lu(97to971i). (2.7)
Remark 2.3 If we define 9% = g (97Ioo 971io), ^1 = gL (SOToi min)
and mV mi'0 rni[, then in general, 971q ^ 9710 and mi\ ^ 971!, but it follows
from (2.7) that 971' = 971.
Definition 2.4 ([11], [24]). We say that a pair of subspaces IF and g are in
generic position if the four intersections 97loo,371oi)lio o-nd 971n, defined by
(2.4), are trivial, i.e. consist of the zero element only.
14

Remark 2.5 It follows directly from Definition 2-4 that the subspaces IF and Q
are in generic position if and only if any of the pairs F and Gx, or FL and G,
or F1 and G1 is in generic position.
Lemma 2.6 Let F and Q be the subspaces ofH, and a subspace Tl, defined by
(2.5), (2.6), be non-trivial. Then PmF and PmG, as the subspaces ofTl, are in
generic position.
Proof: Let us first mention that PmF = 9Jl0 = F 0 (3Jt00 9Jt01), PmF1- =
Tli = Fx (HJlio 9Jtn), PmG = TV0, and PmGx = where
2K(, = Se(mtoomt10), 2rt'1 = 0-Le(2rtOi9Jt11). (2.8)
Next, we see that Tl0 X Tl\, Tlo 97li = and 9JIq X Tt[, Tl'Q
Tl'i Tl. If we assume that any pair of these four subspaces has a nontrivial
intersection, then it evidently is a subspace of one of the four subspaces 97loo,
SDToi, 9Jtio or 27tn. This contradiction shows that all intersections are trivial,
and consequently, PmF and PmG are in generic position (with respect to SPT).
The following theorem characterizes the projectors onto subspaces that are
in generic position.
Theorem 2.7 ([24])- If F and G are subspaces in generic position in a Hilbert
space TL, with respective orthogonal projectors P? and Pg, then there exists a
Hilbert space TL, and there exist positive selfadjoint commuting contractions S
and C on TL, with S2 + C2 = 1 and T(S) = T(C) = {0}, such that PT and Pg
15

are unitarily equivalent to
respectively.
(l 0
1 0
and
f c2 cs
\cs S')
(2.9)
Using Theorem 2.7 it is proved (see [65]) that the projectors P? and Pg
admit the following representation corresponding to the decomposition (2.4),
(2.7):
where
/
PF = //00
/ 0
0 0
Pe = /0/0W*
^ c2 cs^
yCS S2 j
W,
w =
I 0
0 w
(2.10)
W : 9Jli > StTlo is an unitary operator and S = y/PpPg 1^, C = \fPrPg\vno-
The representations (2.10) show that all five parts of the space H are invari-
ant for both operators Pjr, Pg and also for all operators, which are represented
as polynomials of these two operators. Moreover, we can see, for instance, from
the first equality of (2.10) that 9710o and -DDToi are the eigenspaces of the operator
Pjf corresponding to the eigenvalue 1 and 97ti0,9Jln are the eigenspaces of the
same operator, corresponding to the eigenvalue 0.
16

Lemma 2.8 The four operators Pjr, Pg, Pp and Pg commute if and only if
the spectra E(PjrPg), E(PjrPg.L), E(PjrPg) and T,(PjrPg) are the subsets of
{0}U{1}.
Proof: Assume that P? and Pg commute (which means that all four projectors
commute). Then
PpPg = Pj7 net PfPg^ P^ngi Pr-Pg = P^ngt PfPq = Pr-rg-
In its turn, these equalities give then that all of the sets E(P^Pg), E(PjrPg),
E(Pjrj_Pg) and E(PFPex) are the subsets of {0} U {1}.
Assume now that all four spectra are subsets of {0}U{1} but PfPg PgPr-
In this case 371 = {0}, and we can represent the space H as Tt = 9Jtoo 91toi
3Hio 37ln. In this case it can be easily checked that
PjrPgU = PgPfU = U
for Vu G SDToo and
PjrPgU = PgPyrU = 0
for Vu G 9Jl0i Tio 971 n since this equality holds separately for all three
subspaces, that is for Vu 97loi; Vu G 9ftio, Vu G 97tn-
Lemma 2.9 The operators P^Pglm, PpPg^lm, P^Pg\m, and PjrPgx\on are
Proof: Direct calculations using (2.10) show that the operator block represen-
tations of the operators considered here are:
/ C2 0
P?Pg |an
\ 0 0
PrPg^lm
0
0
(2.11)
17

and

PpPg\m
0 0
0 W*S2W
Pf-lPq- |an
0 0
0 W*C2W
(2.12)
Since S2 and consequently, also C2 I S2 are positive selfadjoint contractions,
and W is unitary operator, these representations show that all four considered
Similarly, the operators PgP?|gjt, PgPr\m, PgP?\m, and PgP?j-|ot are
Lemma 2.10 The point spectra of the restrictions of operators P^PgPjr and
PjrPgPjr to the sub space 9710 and the restrictions of the operators P:FPgP:F,
and Pjr Pgi. Pjtl to the subspace 9711, respectively do not contain ones and zeros,
that is 0,1 ^ Hp{PjrPglajib), 0,1^ T,p(PjrPg\m0) and 0,1^ Hp(PjrPg
0,1 ^ T,p(P:FxPg\tm1).
Proof: Using (2.11) and (2.12), we have P?Pg|ajib = C2 and P^Pq^i = S2.
But then, if we assume that 1 G T,p(PyrPg\ get 91(5) 7^ {0}, which contradicts Theorem 2.7. Next, since C2 + S2 = /, if
1 is an eigenvalue of S, then 0 is an eigenvalue of C, and vice versa. Based on
the last fact, we see also that if 0 G Ep(PjrPg|gji0), or 1 G Ep(PFi.Pe|OTl), then
91(C) ^ {0}. These contradictions to the properties of the operators S and C,
given in Theorem 2.7, show that the assertion is proved.
Corollary 2.11 The point spectra of the restrictions of operators PgP?Pg and
PgPpA-Pg to the subspace 971q and the restrictions of the operators Pg PpPg^
and Pgj.Ppi.Pgi. to the subspace 971 j, respectively do not contain ones and zeros,
18

that is 0,1 ^ Y.v{PgPj0,1^ H^PgPjr/m/), 0,1 ^ T,p(PgPyrl^) and
0,1 ^ Ep(PgPFx|an').
Remark 2.12 (see Remark 4-^7, P ^28, [37]). The five parts are unstable
with respect to perturbations of T and/or Q and thus cannot be used in the
perturbation analysis.
The following lemma contains simple but useful facts about the first four
parts of the space. All assertions assume that corresponding subspaces Too,
toi, bo and In defined by (2.4) are nontrivial.
Lemma 2.13 The following assertions are true:
The subspace l00 is an eigenspace of the operators PjrPg\-F and PgP^\g,
corresponding to the eigenvalue one, and is an eigenspace of the operators
P?Pg\?r and PgPp\g, corresponding to the zero eigenvalue;
The subspace T01 is an eigenspace of the operators PjrPg |jr and Pg Pp\g,
corresponding to the eigenvalue one, and is an eigenspace of the operators
PpPg\jr and Pg^Pjr/gr, corresponding to the zero eigenvalue;
The subspace T10 is an eigenspace of the operators P:FPg\-F and PgP?-\g,
corresponding to an eigenvalue one, and is an eigenspace of the operators
and PgPjr\g, corresponding to the zero eigenvalue;
The subspace In is an eigenspace of the operators PjrPg ^ and
PgPfx\g, corresponding to the eigenvalue one, and is an eigenspace of
the operators P?Pg\jr. and PgP?-\g, corresponding to the zero eigen-
value.
19

2.3 Isometries Between Subspaces
In this section we study two important mappings the isometries of Sz.-Nagy
[62] and of Kato [37], Davis and Kahan [12]. The second mapping is the unitary
version of the first one and is called a direct rotation. It maps the subspaces T
and TL onto the subspaces Q and G1, respectively.
We assume that the inequality
IIPr -Pell < 1 (2.13)
holds. As we can see below (Lemma 2.36), this property is equivalent to a
property that a set of angles between T and Q does not contain a right angle.
The Sz.-Nagys partial isometry [62] for a given subspaces T and Q is defined
by following equality
W = PgA-1/2Pf, (2.14)
where
A = I P? + PtPqPt = A*. (2.15)
It follows from the assumption (2.13) that \\Py:(Pg Pjf)-P.f|| < 1 holds, and
consequently
A = I + PAPg ~ Pr)Pr > (1 IIPt~ Pg\\)I > 0-
Thus A~XP exists and is positive, bounded and selfadjoint. Furthermore, Pjr
commutes with A and, consequently with A-1/2 too. Based on these facts we
get [62] that
W*W = Pjr, WW* = Pg. (2.16)
20

The relations (2.16) imply that IT is a partial isometry (see, e.g., [73], p.85): its
restriction to the subspace T is an isometry, that is it leaves an inner product
of the elements of T unchanged, and WuL = 0, for Vir1 E T .
The mapping, constructed by Kato [37], Davis and Kahan [12], is a unitary
mapping, which coincides with the mapping given by (2.14) on
w = [PgPr + (I Pg)(I Pr)](I i!)-1/2, (2.17)
where R = (Pjr Pg)2. It is easy to show that the factors in the right-hand side
of W commute. Further, WW* = (/ R)~l^2{I R)(I i?)-1/2 = I, (see [58])
so W is unitary.
Now let us explore the relations between the partial isometries in the polar
decomposition of operators and (partial) isometries considered above.
Definition 2.14 (Definition 3.10, p. 242, [10]). A partial isometry is an oper-
ator W E B(H) such thatfor'iu E ^(TT)1, ||Wu|| = ||u||. The subspace Vl(W)
is called the initial space of W and the subspace 9t(lT) is called the final space
ofW.
The following fact is well known:
Lemma 2.15 (Theorem 3.11, p. 242, [10]). If T E B(H), then there is a
partial isometry W with yt(T) as its initial space and 91(T) as its final space
such that T = IT|T|. Moreover, if T = UQ, where Q > 0 and U is partial
isometry with 01(17) = 91(Â£?), then Q = |T| and U = W.
Theorem 2.16 For both choices ofW = W and W = W, where W and W are
defined by (2.14) and (2.17), respectively PgP? = W\PgPjr\ holds.
21

Proof: Denote B = P^Pg. Then B* = PgPr, BB* = P^PgP^r and B*B =
PgPfPg. If we denote S = (BB*)1/2, then considering the polar decomposition
(see [73], [37]), B* can be represented in the following form B* = WS (for B we
have B = SW*), where W is a partial isometry with initial domain P and final
domain Q. Note also that B*B = WSSW* = W(BB*)W*.
First let us consider the case W = W, where W is defined by (2.14). We
have:
WS = PgA-X'2PTyJPTPgPT = PgP?A~ll^PTPgPf. (2.18)
But the last multiplier in (2.18) is zero on PL and A~~1/2 = (PjrPgPjr)~1'2 on P.
This means that WS = PgPp = B*. From this last equality we conclude that
the partial isometry used in the polar decomposition is the same as Sz-Nagys
partial isometry, given by (2.14), which maps isometrically T onto Q.
In the case of W = W, where W is given by the relations (2.17) we have:
WS = (PgPr + (I- Pg)(I PrW ~ R)-1/2VPrPgPr. (2.19)
But analogous to the above arguments we see that WSu1- = 0 for u1 P1.
Next, I R = PyrPgPj: on T, and we conclude from the equality (2.19) that
WS = PgPjr = B*. And again, we see that the isometry used in the polar
decomposition can be given by (2.17).
Let us now consider a question of the unitarily equivalence of the operators
PfPgPr and PgPjrPg.
Theorem 2.17 ([43])- Let T and Q be subspaces of H. Then there exists a
unitary operator W G B(T4) such that
PTPgPT = W*PgPTPgW. (2.20)
22

Proof: Denote T = PgPjr, then T* P^Pg. Using Lemma 2.15 and the
fact that 91(T)J~ = 91(T*), we conclude that there exists a partial isometry
U : Wt*) -* WT), such that T = U^PTPgPT holds.
It is known (see, e.g., Theorem 1, p. 126, [54]) that a partial isometry U
has a unitary extension if and only if ((P))1 and (91(U))-L are isomorphic
(Definition 5.1, p. 19, [10]). Since D(P) = 91(T*) and 91(Â£/) = 91(T), we
conclude that for existence of the unitary extension of U it is necessary and
sufficient 91 (T*) and 91 (T) to be isomorphic. This means, that 91 (P^Pg) and
91 (PgPyr) should be isomorphic. Using the decompositions (2.10), we have
91 (PyrPg) = 97loi 9Hio Win 91(PjrPg|in),
91 (PgPj?) = 9Jl0i 9Hio Win yt(PgPr\fm)-
These relations show that to prove that 91 (PjrPg?) and 91(PgP^), it suffices
to show that ^{P^Pglm) and ^l{PgPr\m) are isomorphic, which follows from
Theorem 1, [24]. Finally, we conclude that there exists a unitary extension of
U. Denote it by W and we have
PgPT = Wy/PrPgPr, P?Pg = y/PfPgPfW*.
Multiplying these equalities we obtain
PgPjrPg = WPTPgPTW\
which is equivalent to (2.20), since W is unitary.
23

2.4 Definitions and Properties of the Angles
In this section, we define the angles from one subspace to another one, and
angles between subspaces. Next, we investigate the properties of the (sets of)
angles, such as the relationships connecting the angles between the subspaces
and and their orthogonal complements. The idea of definition of angles between
infinite dimensional subspaces using the operator spectral theory appears in [26].
Definition 2.18 A set
Q{F, G) = {8 0 = arccos(a),
is called the set of angles from the
Definition 2.19 A set
(F, G) = e{F, G) n e(G, F) (2.22)
is called the set of angles between the subspaces F and G
Remark 2.20 In general Q(F,G) ^ {G,F), that is the set-valued function
Q(F, G) is non-symmetric, but &(F,G) is a symmetric function.
Definition 2.21 QP(F,G) = {0 0(F, G) ' cos2(0) is an eigenvalue of
P?Pg\r}-
Definition 2.22 9P{F,G) = %{F,G) n &P(G,F).
The following technical lemma describes a relationship between the spectra
of operators PjrPg\jr and P?Pg.\?.
0, <72eE(PfPe|r)} C [0,|] (2.21)
subspace IF to the subspace G
24

Lemma 2.23 If a2 G T,(PjrPg\r) then /P 1 a2 G E(PjrPg\jr) and vice
versa, that is if p? G T,(PjrPg |jr) then a2 = 1 p2 G E(PjrPg|:F)-
Proof: The proof immediately follows from the equality
-fV-Pg|;F + P^Pg-t-lr = PrW = I\f,
and the spectral mapping theorem (see, e.g., [77], Corollary 1, p. 227), which
states that if / is a complex-valued function, holomorphic in some neighborhood
of E(P), then E(/(T)) = /(Â£(T)). In our case f(T) =I-T. m
Using definition 2.21 in Lemma 2.23 we can interpret this result in terms of
the angles.
Theorem 2.24 Q(P,Q) = f 0(P,&X).
Interchanging the subspaces gives Q(Q,P) = f Q(G,IP1)-
Using Theorem 2.24 we can introduce equivalent definition of the angles
using the sines.
Definition 2.25 The set
e(F,G) = {6 : 0 = arcsin(/i), p>0 p2 G E(P^Pg|^)} C [0, |] (2.23)
is called the set of angles from the subspace T to the subspace Q.
Theorem 2.26
e(5,^)\{f} = 8(^,5) \{I}. (2.24)
Proof: The assertion is a particular case of a more general fact: if A, B G B(H),
then non-zero elements of the spectra of operators AB and BA are the same (see,
25

e.g., [64], Chapter 10, Exercise 2, or [10], Exercise 7, p. 199). Here A = PjrPg
and B = PgP?.
Let us investigate where the non-symmetry of (P, Q) comes from. Using
decompositions (2.10) for the projectors P? and Pg corresponding to the five
parts of the space H, we get:
P?Pq\r -^OToo Ojmoi PpPg Ian,
PqPtIg -^9%o Oartio PgP?Ian- (2.25)
Next, using Theorem 7.28, p. 208, [73], which states that if T B(7H) is
selfadjoint and M is its invariant subspace, then the restrictions TM, TM are
selfadjoint and E(T) = T,(TM) U H(TM), we have E(PJrPe|:r) = T,(Im00) U
E(0OToi)UE(P^Pe|OT) and E(Pg;P^r|g;) = E(7aHo0)UE(Oot10)^^{PgP^M- These
two equalities show that both sets of angles 0(P, Q) and (^, JP) simultaneously
contain or do not contain the zero angle, but with the right angle the situation
is different. Namely, because of the difference in the second terms of the right-
hand sides, one of these sets may contain a right angle, when the other one does
not.
As we can see, there are eight, in general different sets of non-symmetric
angles for a given pair of subspaces. Let us investigate the relationships between
them. It is sufficient to get these relationships, for instance between 0(P, Q) and
the other seven sets. The relationships between arbitrary pairs of sets of angles
can be easily gotten from these seven relations by interchanging the subspaces.
Two of these seven equalities are already given in Theorem 2.24 and Theorem
2.26. We consider first the angles from one subspace to another since they reveal
26

finer details. We give here all seven relations for completeness. For symmetric
angles we obtain analogous results later in this section.
Corollary 2.27 The following relations hold for any pair of subspaces T and
Q of hi:
i. e(F,g) = z-e(r,g1);
e(e.j-) \{f} = e(^,e) \{f};
s. &(r\g) $${0}u {Â§ = f {B(r,g) \({o}u{Â§})},- l e(Tx,gx) \({o} u{?}) = e(r,g) \({o} u {Â§}),- s. e(e,^-L)\{o} = f-{e(^,e)\{f}},- 6. 0(gL,r) \ {Â§} = f {(^.s) \ {o}},' 7. Q(gx,rx) \ {o} = e(r,g) \ {o}. Proof: The first two assertions axe already proven. To prove the other equalities, we use these two relations. 3. &(r\s) \ ({o} u {f}) = 0(5,^) \ ({0} u {Â§ = {f e(g,r)} \ ({0} U {f}) = {f {0(e, F) \ {|}}} \ {|} = {f Q(F, 5)} \ ({0} U{f = f-{0(JF,5)\({O}U{Â§})} 4. Using the previous equality, we get 0iJ-L ,GX) \ ({0} U {|}) = {| e(F, 5X)} \ ({0} U {f}) = Q(F, g) \ ({0} U {f}); 5. Q(g,Fx) \ {0} = {f 0(5, J-)} \ {0} = f {0(5, .70 \ {f}} = f - {^,e)\{f}}; 27 e. e(s,jr)\{|} = e(Jr,e1)\{|} = {f-e(^,s)}\{f} = f-{e(^,s)\ {o}}; 7. 0(S^)\{O} = {f Q(Q^,T)} \ {0} = i-{e(^,gl)\{|}} = {| (?, e1)} \ {o} = e(r, g) \ {o}. Remark 2.28 Let us explain why only one of all the relationships between the two sets of angles does not require elimination of the zero or right angles. As we have seen in Theorem 2.24, this pair is 6(3, Q) and Q(J-, Qx) (analogously, Q(G,P) and 0(^,^r-L)y). The key to the answer is the fact that the operators, corresponding to these sets of angles, have the same eigenspaces outside of the fifth part of the space Tt. These eigenspaces, as is shown in Lemma 2.13, are QJloo and 97toi They are the eigenspaces of the operator PpPg\jr, corresponding to the eigenvalues 1 and 0, respectively, and the eigenspaces of the operator P?Pg |jr, corresponding to the eigenvalues 0 and 1, respectively. The eigenspaces of other pairs do not match, which causes a possible difference between the spectra. To illustrate the results of Theorem 2.24 and Corollary 2.27, consider the following example, which is similar to that considered in [42]. Let TL be a real Euclidean space R6, T be a subspace of dimension 4, spanned by the columns of a matrix [J4,0]T 6 and Q be the three dimensional subspace, spanned by the columns of a matrix [Di,D2]t where Di and D2 are diagonal matrices, whose diagonal elements are 1 /1 + df and di/yj 1 + df i = 1,2,3, d\ = 0, and d2, are non-zero real numbers. Then it can be easily seen that the following equalities hold: cos(0(J-, Â£)) = {!; 1/y/T+~df; 1/VT+df; 0}; 28 sin((P, Q1')) {1; \j yj\ + dj; 1/y/l + d0}; sin(0(Px, Q)) = {l/i/l + d\] 1/v/l + df}; cos(0(^r-L,^-L)) = {l/y/TTdf; l/y/l + d%}] cos(0(Â£,P)) = {1; 1/x/TTdf; l/y/\ +^}; sin(0(S,Px)) = {i/>/rTdf; i/\A + ^; 0}; sin(0(^-L.^')) = {1; l/v/l + df; l/-y/l + dl}; cos(0(^1,^-L)) = {1/\/l + d\\ l/y/l+(%\ 0}. Corollary 2.29 0(P,0) \ {|} = 0(P,0) \ {f} = 0(0, P) \ {f}. Lemma 2.30 Let P and Q be subspaces of H, and subspace DJI, be defined by (2.5), (2.6), be non-trivial. Then 0(P, Â£?)\({0}U{|}) = 0(P*nP, P (ID- Proof: Using (2.25), we have ^P^Pg]?) = E'US(PjrPg|an) and Y>(PgPp\g) = E" U E(PgrPjr|gjt)) where E',E" C {0;1}. But then we get cos(0(P, G)) = E' D Ew D E(7^rPg)oji) 0 E(P E' D E" C {0; 1}, the assertion follows from the last equality. The following lemma describes a trivial case, when the projectors P? and Pg commute. Lemma 2.31 If P?Pg PgPf then all sets of angles 0(P, (/), 0(P,G1), 0(PX,0), (P1,*?1), 0(0,P), (0,PX), 0(0X,P), 0(0X,PX) are subsets of the set {0; |}, that is the only the possible angles between the subspaces P, Q, Px and Q1- are the zero and right angles. 29 Proof: If Pyr and Pg commute, then also all polynomials in Py and Pg com- mute, particularly all four projectors Py, Pg, Py and Pg commute. But then (see, e.g., [2], p. 65, Theorem 1) we have PyPg = Pyng, P^Pg = Pfng, PyPg = Pyrrg, PyPg = Pyng. Then the assertion follows from the fact that the spectrum of orthogonal projector consists of two points zero and one. Let us now consider the relationships between the quantities defined by (2.1), (2.2) and angles, defined in the current section. Lemma 2.32 c0(P,G) = sup{cos(0(^',^))}, (2.26) and c(F,G) = sup { cos (e(P,G) \ {0})}. (2.27) Proof: We have using Theorem 5.35, p.120, [73] about definition of a bounded linear operator by bilinear form: Co(P, G) = sup sup |(lt,u)| = sup sup \(PyU,PgV)\ = ueT veg uen ven IW|=i M=i llll=i IN=i sup sup \{u, PyPgV) \ = ||PjrPjj|| = sup { cos (9(.F, Q))}. uen ven ||u||=l ||v||=l For c{T,G) we have: c{T,G)= sup sup |(tt,v)| = ueTv\imgp vegnirng^ ||u||=l IM|=1 SUp SUp \{PyU,Pgv)\ = SUp SUp \(u, PyPgv)\. uenofxttoo venovRoo uensvoioo venoffioo Nl=i Ml=i IM!=i IN=i 30 Using the decompositions (2.10), we have PpPg =/000 PpPg (2.28) where I and 0 are the identity and zero operators on the corresponding subspaces of H. (2.28) shows that excluding the subspace 9Jl0o from a domain of the supremum excludes 1 from the spectrum of P?Pg and we obtain (2.27). Let us denote by gap(P,Q) a gap (aperture) between subspaces P and Q (see, e.g., [2], [22], [37]) which is defined by gap(F,Q) = ||Pr Ps\\ max{||P?Psi||, ||P9P,||}. (2.29) Theorem 2.33 Let P and Q be subspaces of the Hilbert space Tt. Then min{min{cos2(0(.F, Â£?))}, min{cos2(0(Â£, P))}} = 1 (gap(P,Q))2. Proof: Let us consider both norms in the right-hand side of (2.29) separately. We have using Corollary 2.27: \\PyrPg\\2 = sup \\PfPgxu\\2 = sup {PfPgxU, PjrPgxu) = uÂ£H l|u||=l IMI=1 sup (PgP?Pgu,u) = ||PgxP^|e|| = max{cos2((61,P))} = ueH M=1 max{sin2((Â£?,P))} = 1 min{cos2(0(Â£?,p))}. (2.30) We obtain similarly \\PgPr\\2 = max{cos2(0(P, Â£?))} = 1 min{cos2(0(P, Q))}. (2.31) Equalities (2.30), (2.31) lead to the assertion. 31 Remark 2.34 Corollary 2.29 describes only the relationship between non-right angles from T to Q and from Q to IF. What can we say about the light angles? There are several possibilities: neither 6(F,G) nor Q(G,F) contain a right angle. In this case B(F,G) = B(F,g) = B(G,F); one of the sets of angles contains a right angle as an isolated point, but the other one does not; both of them contain a right angle; in this case the multiplicities of the right angle can be different. The following Theorem gives the relationships between the sets of angles between different pairs of subspaces. Theorem 2.35 For any subspaces F and G of H the following equalities Q) \ ({0} u {^}) = {Â£ b(f,g)} \ ({0} u {Â£, (2.32) Q(F,g) \ {0} = Q{FX,GX) \ {0} (2.33) and e(F, <3X) \ {0} = b(fx, G) \ {0} (2.34) hold. Proof: We have using Theorem 2.24 (F,G) \ ({0} u {|}) = B(F,G) \m u {^}) = {\ C^)} \ ({0} u{^) = - b(f,gx)} \ ({0} u {Â£}). 32 Using the seventh equality of Corollary 2.27 twice first for T and Q, next for Q and J-, and intersecting them gives (2.33). Interchanging Q and Q1- in (2.33) gives (2.34). If T and Q are proper subspaces of H then evidently 0 G 'E(PjrPg), but at the same time 0(.F, Q) may not contain |. Consequently, using Lemma 2.1, we can write Â£(PrPg) = cos2((^, g)) U {0}. (2.35) Let us mention that we can also prove equality (2.33) using the following theorem, which describes the relationship between the spectra of the product and difference of two orthogonal projectors. Lemma 2.36 ([57], Theorem 1. See also [44], Lemma 2-4)- For any pair of orthogonal projectors P? and Pg on Tt the spectrum of the product PpPg lies in the interval [0,1] and S(PS-P^)\({-1}U{0}U{1}) = {(1-<72)1/2 : <72 Â£ E(P^PS)\({0}U{1})}. (2.36) Using Lemma 2.36 and equation (2.35) the following theorem and corollary are proven: Theorem 2.37 Â£(P^ Pg) \ ({1} U {0} U {1}) = sin(0(^r, Q)) \ ({1} U {0}U{1}). The multiplicity of an eigenvalue 1 in T,(Pg P?) is equal to dimQJtio, multiplicity of an eigenvalue 1 is equal to dimQJtoi, and multiplicity of an eigenvalue 0 is equal to dim 97l0o + dim 9Jtu, where 9ft0o> QJtoi, and Wlu are defined by (2.4). 33 Proof: (2.36) is proven in [57]. To obtain the results about the multiplicity of eigenvalues 1, 1 and 0, it suffices to make use the decomposition of these projectors in five parts, given by (2.10). Corollary 2.38 X((Pr Pg)2) \ ({0} U {1}) = sin2(0(.P, Q)) \ ({0} U {1}). In many applications, such as domain decomposition algorithms, existence of the information about the distribution of spectrum of the sum of projectors is important. The results about spectra of sums of projectors can be found e.g., in [7] (See also [72], Theorem, p. 298). Using Corollary 4.9, p. 86, [7], we can formulate the following result in terms of the angles between subspaces. Theorem 2.39 For any nontrivial pair of orthogonal projectors Pp and Pg on 7i the spectrum of the sum P'? + Pg with possible exception of point 0 lies in the interval [1 ||P^-Pg||, 1 + ||P^Pg||] and + Pg) \ ({0} u {1}) = {1 cos(0(P, 0))} \ ({0} U {1}). (2.37) 2.5 Angles Between the Subspaces in Generic Position The main observation for subspaces in the fifth part is that the projections of initial subspaces onto the fifth part are in generic position. This fact allows us to conclude that the zero and right angles can not belong to the set of angles between these new subspaces (and also to the set of angles from one subspace to another one) as isolated angles. This simplifies the assumptions of several propositions given above in this chapter. Lemma 2.40 Let the subspaces T and Q be in generic position. Then 1 ^ Ttp(P^Pg\^) and 0 ^ T,p(P^Pg\^). 34 Proof: Assume first that 1 Â£ Tv{P^Pg\jr). Then we have PjrPg\jru = u for some unit u Â£ T. From the last equality we conclude that u Â£ Q, since otherwise we would have \\Pgu\\ < ||it|| = 1, and consequently 1 = ||it|| = \\P^Pgu\\ < ||P^w|| < 1. This means that u Â£ Q, and consequently T tl G ^ {0}. But this contradicts our assumption about the generic position, that is 1 ^ Ep(PjrPg\:F). Now assume that second assertion does not hold. Then we have PpPg\fU = 0 for some nonzero u Â£ T. Next, if Pgu = 0, then T D Q1- ^ {0}. If Pgu ^ 0, denote v = Pgu Â£ Q and we get P^v = 0. But this means that Jn G1 {0}. The contradiction in both cases shows that 0 ^ Ep(PjrPg|jr). Corollary 2.41 If the subspaces T and Q are in generic position, then 0 ^ Op(f,G) and | Â£ Op(P,G)- Proof: It sufficies to mention that the definitions of the generic position and angles between subspaces are symmetric with respect to the subspaces. Theorem 2.42 Let a subspace SOT defined by (2.5) and (2.6) be non-trivial. Then WS)\({o}u{^}) = ep(PmF,pmg). (2.38) Proof: By Lemma 2.30, ep(P, G) \ ep(Pmf, PmG) c {0; |}. (2.39) But, following Lemma 2.10, the set of angles 0p(Pgjt-P) PmG) does not contain zero or right angles. Consequently, (2.38) follows from (2.39). The following theorem gives the sufficient conditions for two subspaces to be in generic position and describes additional relationships between different 35 sets of angles. Theorem 2.43 The subspaces T and Q ofTi are in generic position if and only Proof: Let 0, | ^ QP(iF, g) U Qp(g,lF). If we assume that F and g are not in generic position, then some of the four subspaces 97I00, 9DToi, 9Kio, SDTn given by (2.4) is nontrivial. Then we have zero (or right) angle in above given set(s) of angles. This is a contradiction to the assumptions of the theorem, and we conclude that T and g are in generic position. Assume now that T and g are in generic position. If 0 G 0p(^r, g)UBp(g, J-) then the subspace 37l00 is nontrivial, which contradicts the assumptions. If | G p(f, g)UQp(g,iF) then one of the subspaces 9Jloi> 9ftio is nontrivial, which also contradicts the assumptions. The absence of the zero and right angles in the set 0(J-,g) U Q(g,iF) is only a sufficient condition for two subspaces J- and g to be in generic position. But, evidently it is not a necessary condition. If T and Q are in generic position, then no set of angles, contains zero or right angle as the element of 0p, but these angles may belong to the set of continuous angles, which we define now. Definition 2.44 Qc{T,g) = 0(JF, g) \ Qp(T,g). Let us call c(Jr, (?) the set of continuous angles from the subspace T to the subspace g. Remark 2.45 This definition of the continuous angles is based on the definition of the continuous spectrum of self adjoint operator (see, e.g., [16], Definition 1, 36 p. 902). Let us mention also that since the spectrum of a selfadjoint operator con- sists only of the eigenvalues and continuous spectrum (the residual spectrum of selfadjoint operator is void), by Definition 2.21 and Definition 2.44 we have defined all possible angles from the subspace T to the subspace Q. Theorem 2.46 Let 0 = 0 or 6 = f. If 0 Â£ QC(!F,G), then: 1. 0 belongs also to the sets QC(G,F) and QC(G,J7); 2. | 0 belongs to the sets QC{!F,GL), QC{IFL,Q), Qc(G,3r'L) and 0C(^1, IF). Proof: Since 0 6 0C(P> Q) and 0 is not an eigenvalue, there exists a sequence {0*}^ C Q(P,G) \ ({0} U {f}) such that limi_>oo0j = 0. But then, from Theorem 2.24 and Corollary 2.27 it follows that is a subset of all three sets, listed in Assertion 1, and {f 9i}\ is a subset of all four sets, listed in Assertion 2. It suffices only to mention that all sets of angles, considered here, are com- pact sets. Prom Theorem 2.43 and Theorem 2.46 we obtain the following Corollary 2.47 If the subspaces J- and G are in generic position, then = X(PgPr\g), and 0(Jr,G) = e(G,F). 37 Remark 2.48 Using Theorem 2.f3 and Theorem 2-46, we conclude that if the subspaces F and G are in generic position, then the assertions of Theorem 2.26, Corollary 2.27, Corollary 2.29, Lemma 2.30, Theorem 2.35 hold without exclud- ing the zero and right angles, and the assertions of Lemma 2.36, Theorem 2.37, Corollary 2.38, Theorem 2.39 hold without excluding {0}, {1}. Let us now consider the relationships between the quantities defined by (2.1), (2.2) and angles, defined in the previus section in the case, when F and Q are in generic position. Corollary 2.49 (of Lemma 2.32 ). If F and Q are in generic position, then c0(F, G) = c(F, G) = sup { cos (G(F, G))} (2.40) Proof: It suffices to mention that 9Hoo = {0} since IF and Q are in generic position, where 97l0o is defined by (2.4). 2.6 Principal Vectors In this section, we define the principal vectors and investigate their main properties. Several cases are investgated, when the properties of principal vec- tors are similar to those of principal vectors in the finite dimensional case. Based on a given pair of principal vectors, the pairs of principal vectors are given for different pairs of subspaces. Special attention is paid to the cases, when the zero or right angle is present as an isolated angle. Definition 2.50 Normalized vectors u = u(6) F and v = v(6) e G form a pair of principal vectors for subspaces F and G corresponding to the angle 0 Â£ Q(F, G), if the equalities PjrV = cos(0)u, PgU cos {0)v (2.41) 38 hold. Definition 2.51 Assume 8 Â£ Qp(lF, G) \ {f}- Then, the multiplicity o/cos2(#) as an eigenvalue of the operator P^Pg\r, is called the multiplicity of an angle 6. The following auxiliary lemma shows the correctness of last definition and is useful in the rest of this section. Lemma 2.52 If u Â£ IF is an eigenvector of an operator Pj?Pg\jr corresponding to the eigenvalue cos2 (8), 8 7^ f, then v = Pgu Â£ Q is an eigenvector of an operator PgPj?\g corresponding to the same eigenvalue. The multiplicities of cos2(8), as an eigenvalue of the operators Pj?Pg\jr and PgP^\g} are the same. Proof: First let us notice that P?Pg\r is a nonnegative operator, PfPgW > d, since P?PgPf = (PgP^YPgP^, and so is PgP?\g > 0. By the lemma assumption we have Pgu ^ 0 and PrPg\fU = cos 2(8)u. (2.42) Then we have: (PqPAg)Pqu = Pg(P?Pg\Fu) = cos 2(8)Pgu. (2.43) It is easy to show that the converse' relation is also true: if v Â£ Q is an eigenvector of an operator PgPjr\g, corresponding to a non-zero eigenvalue, then Pf v Â£ IF is an eigenvector of the operator Pj^Pg|^, corresponding to the same eigenvalue. To show that the multiplicities of cos2 (8) > 0 as an eigenvalue of oper- ators P?Pg\p and PgP?\g are the same, it suffices to mention the following: 39 if u',u" G T form a pair of orthonormal eigenvectors of an operator PjrPg\jr, corresponding to cos2(0), then Pgu1 / cos(6) and Pgu"/cos(6) form a pair of or- thonormal eigenvectors of PgPp\g corresponding to the same eigenvalue and vice versa. This means that there is a one-to-one correspondence between the orthonormal sytems of eigenvectors of P^Pg]? and PgPj?\g corresponding to the given eigenvalue i.e. the multiplicities are the same. Remark 2.53 The equlity of multiplicities of the nonzero eigenvalues of the operators P?Pg\? and PgP?\g can also be deduced from Theorem 2.17. Lemma 2.54 Ifu T and v Q form a pair of principal vectors for subspaces T and Q corresponding to the angle 6 G S(1F,Q), then (u,v) = cos(8). Proof: Using Definition 2.50 we have (u.v) = (Pjru,v) = (u, Pjrv) = (u,cos(0)u) = cos(), (2.44) since u is normalized. Lemma 2.55 IfuEp and v G Q form a pair of principal vectors for subspaces T and Q corresponding to the angle 6 G Q(P,Q), then u and v are the eigen- vectors of the operators P?Pg\p and PgP?\g, respectively, corresponding to the eigenvalue cos2(0). Proof: Since the assertion is symmetric with respect to u and v, let us prove it only for u. We have: PjrPg\jrU = Pf(PgU) = COS (9)PfV = COS 2(#)u. 40 Theorem 2.56 Let 9 Â£ (J7, Q) \ {|} and u Â£ T be an eigenvector of the operator PjrPg\jr, corresponding to the eigenvalue cos2(9), ||u|| = 1. Then there exists a unique eigenvector v Â£ Q, ||i>|| = 1, of the operator PgPjr\s, correspond- ing to the same eigenvalue, such that u and v form a pair of principal vectors, corresponding to the angle 6. Proof: Let v = (l/a)Pgu, where a = cos(9). Then v Â£ Q, by Lemma 2.6 v 7^ 0 and v is an eigenvector of PgP?\g, corresponding to the same eigenvalue a2. We have P?v = Pr((l/a)Pgu) = (l/a)PjrPgPjru = cru. Thus Pgu av and P^v = ou, that is both of the equalities (2.41) hold. Also ||t>||2 = (v,v) = (1 /o)(v,Pgu) = (l/o)(Pgv,u) = (u,it) = 1. This means that u and v form a pair of principal vectors corresponding to the angle 9. Now let us show that v is unique. If there exist two different vectors V\ and v2 which with u form two diferent pairs of principal vectors, then we have Pgu = av\ and Pgu = av2. Subtracting these last two equalities we get er(iq v2) = 0. But a ^ 0 by the assumptions of theorem. Consequently vx = v2. It follows from Lemma 2.55 and Theorem 2.56 that Theorem 2.57 There exists a pair of principal vectors for subspaces T and Q corresponding to a given angle 9 Â£ (.T7, Â£)\{f} if and only if 9 Â£ &P(F, f?)\{f } Remark 2.58 If 9 Â£ p(^7, G) \ {f} is an angle of multiplicity one, then the subspaces span{u} and span{v}, corresponding to 9 are unique. If 9 Â£ 41 Qp{F,G) \ {f} *s multiple (of finite or infinite multiplicity) angle, then we have more then one pair of principal vectors, but each principal vector is defined cor- responding to the other vector uniquely. Remark 2.59 If 9 in Theorem 2.56 then the vector v may not be unique. Assume for example, Ti = R3, T span{u} and Q = span{vi,V2}, where u = (1,0,0)T, Vi = (0,1, Of and v^ = (0,0,1)T. Then we have P?v\ = 0 u and P?v2 = 0 u. Lemma 2.60 Let 9,(p G Qp{P,G) \ {f}, and u(9),u() G T, v(9),v() G Q be corresponding principal vectors. Then, if 9 u{(f>), v(9) and v(), u(9) and v(4>) are orthogonal. If 9 = , then for a given u{9), there exists v(9), such that (u(9),v(9)) = cos(9) holds. Proof: Orthogonality of the first two pairs follows from the fact that eigenvectors of a selfadjoint operator corresponding to different eigenvalues, are orthogonal (see e.g., [29]). Next, we have: M0),v(0)) = (Pfu(9),v()) = coa()(u(e),u()), but the last inner product is equal to 0 if 9 ^ and to cos(0) if 6 = and both vectors in the last inner product are the same. Corollary 2.61 A pair of principal vectors u, v spans a two dimensional sub- space, invariant with respect to the orthogonal projectors P>, Pg, I P? and I Pg. Such subspaces, corresponding to different 9 s are mutually orthogonal. The following theorem describes relationships between the angles of different pairs of subspaces. 42 Theorem 2.62 Let u and v form a pair of principal vectors for subspaces P and Q, corresponding to the angle 6 G Qp(P, Q) \ ({0} U {f }) Define u = (v cos {9)u)/sin(), v = (u cos {9)v)/sin(0). (2.45) Then u, v are the principal vectors for subspaces P1 and Q, corresponding to the angle | 9; u, v are the principal vectors for subspaces P and G, corresponding to the angle \ 9; u, v are the principal vectors for subspaces PL andQ-1, corresponding to the angle 9. Proof: Denote a = cos(9) and p = sin(0). Evidently u and Pjr(I Pg)PjrU = PjrU PfPgPjrU = (1 Pg(I Pjr)PgV = PgV PgPjrPgV = (1 (T2)v [Pv, (I Pf)Pg(I PjTjUx = {l/p){v au- a2(v au)) p2u. Similarly, (/ Pc)PA1 ~ pg)v-L = (I Pjr){I ~ Pg){I ~ Pr)u- = (I Pg)(I PT){I Pg)v = a2v. 43 Thus. PgUx = fiv, (/ Pf)v = jUWx; PfV = fm, (I Pg)u = HVx, (I P?)(-v) = au, (/ Pg)u= ct(-Uj_). Remark 2.63 It is easy to see that if 6 = 0, then the formulae (2.45) are meaningless. If 9 = then the same formulae give nothing new but the same vectors u and v. Evidently, this does not necessarily mean that we have no principal vectors between the pairs of subspaces, corresponding to the cases 9 = 0 and 6 = |. The following lemma gives more details about the cases discussed in the last remark. Lemma 2.64 If 9 = 0 G BP(IF,Q), then the intersection IF HQ is non-trivial, and if 9 = | Â£ 0P{IF, Q) then both of the intersections T HQ1- or F1 n Q are non-trivial. Proof: Consider first the case 9 = 0 QP(F, Q). Then by Theorem 2.57 there exist a pair of pricipal vectors u, v of subspaces F and Q corresponding to this angle, i.e. to the eigenvalue 1, and we have the equalities P?v = u and Pgu = v. Prom, for instance, the last equality we conclude that u G Q, since otherwise we would have 1 = ||u|| = ||Pgu|| < ||u|| = 1, which is a contradiction. This means that u Â£ Q, and consequently F fl Q ^ {0}. 44 Consider now the case 6 = | G &p{P, Q). Since 0 is an eigenvalue of the operator PjrPg\? we have P^Pgu = 0 for u E P, u ^ 0. Then we have (P^Pgu, u) = 0, which leads to (Pgu, u) = 0. Using the property Pg2 = Pg we obtain (Fgu, Pgt/) = 0, from where we conclude that Pgu = 0. This means that u e Qx, that is u Â£ P n QL. The proof of PL H Q ^ {0} is comletely similar. Corollary 2.65 Under the assumptions of Theorem 2.62, sin2(0) is an eigen- value of the operators Pj^Pgs.\t, PgPj?\g, and Pgj.Pf\g; cos 2(9) is an eigenvalue of the operators P:FPg\jr and PgPT.\g. The corresponding eigenvectors are, respectively u, v, uj_, vj_, u and v_. Corollary 2.66 If 6 G Qp(P, Q) \ ({0} U {|}), then: | -0ep(Fx,g) ; Â§ -deep(P,GL) ; 6 e Theorem 2.67 Under the assumptions of Theorem 2.62, sin2(0) is an eigen- value of the operator (P? Pg)2 with corresponding eigenvectors u + v and u v, where the pair u,v is the pair of principal vectors of subspaces P and Q, corresponding to the angle 8. Proof: We have (Pt Pc? = PAI ~ Pc) + Pei1 ~ PA- (2-46) 45 For each term of the sum on right-hand side we have, based on the Theorem 2.62, Pr{I Pg)u = ^ sin2(#)u (2.47) II Â£ 1 sin2(^)n. (2.48) From the equalities (2.47) and (2.48) it is easy to see that [Pjr(7 Pg) + Pg(I Pr)](u v) = sin2(0)(u v), which based on (2.46) proves the assertion. Let us now formulate one of the results about the eigenpairs of the sum of projectors which are easily constructed using pairs of principal vectors. Theorem 2.68 Let u and v form a pair of principal vectors for subspaces T and Q, corresponding to the angle 6 Op{P, Q) \ ({0} U {f}). Then u + v and u v are the eigenvectors of the operator Pjr + Pg, corresponding to the eigenvalues 1 + cos(0) and 1 cos(0), respectively. Proof: We have (iV + Pg)(u + v) u + cos(6)u + cos(0)i> + v = (1 4- cos(@))(u 4- v), and (PyT + Pg)(u v)=U COS (6)u + COS (9)v V = (1 COs(#))(u V). Remark 2.69 If 9 = 0 e G), then using Lemma 2.64, we can conclude, that T and Q have a nontrivial intersection and, consequently, any nonzero 46 vector, belonging to P V\Q, is an eigenvector of the operator Pjr + Pg, corre- sponding to the eigenvalue 2. Also, using the same lemma, we conclude that if 0 = | E Op(P, G), then at least one of the intersections JF n Q1- or Px n Q is non-trivial. If u Â£ P fl Q1, then PyrU U, PgU U. From the last equality we have Pgu = 0, and by adding this equality to the first of the above two, we get (P? + Pg)u = u. This means that u is an eigenvector of the operator P? + Pg, corresponding to the eigenvalue 1. The same conclusion can be made if PL fl Q is non-trivial. Remark 2.TO Using Theorem 2.43, we conclude that if the subspaces P and Q are in generic position, then the assertions of Lemma 2.52, Theorem 2.56, Theorem 2.57, Lemma 2.60, Theorem 2.62, Corollary 2.66, Theorem 2.68 hold without excluding the zero and right angles. 2.7 Principal Subspaces In this section, we generalize the definition of a pair of principal vectors to a pair of principal subspaces. We show that the set of angles between principal subspaces consists of one point. We also show that the principal subspaces, corresponding to a non-right angle, have equal dimensions. Using a given pair of principal subspaces of P, Q, the pairs of principal subspaces are constructed for different pairs of subspaces, such as T and Qx, P1 and Q, and Q1. Definition 2.71 A pair of subspaces U C T, VC Q is called a pair of prin- cipal subspaces for subspaces P and Q corresponding to the angle 9 Q(P,Q), 47 if the equalities PrPvW = cos 2{9)PU, PgPu\g = cos 2(8)PV (2.49) hold. Let us prove the following auxiliary assertion first. Lemma 2.72 Let U C T be an eigenspace of the operator Pj^Pg\jr correspond- ing to the eigenvalue cos2(), where 9 Â£ Qp(P, Q) \ {Â§}. Then V = PgU is an eigenspace of the operator PgPj\g corresponding to the same eigenvalue, and vice versa: IfVCQ is an eigenspace of the operator PgPp\g corresponding to cos2() 7^ 0, thenU P?V is an eigenspace of the operator P^Pg\jr correspond- ing to the same eigenvalue. Proof: If v Â£ V = PgU, then v = Pgu,u Â£ U. But then PgPj?\gv = Remark 2.73 IfU C T is an eigenspace of the operator PjrPg\T corresponding to the eigenvalue 0, then V = PgU {0}. We now consider the relation between orthoprojectors on the eigenspaces. Lemma 2.74 Let U C T be an eigenspace of an operator PfPg\p, correspond- Pg(PyrPg\:Fu) = cos2(9)v. The second part is analogous. ing to the eigenvalue cos2(9), where 9 Â£ Qp(P,G) \ {|}, Py be an orthogonal projector onto the subspace U, and V = PgU. Then the operator (2.50) is an orthogonal projector onto the subspace V. 48 Proof: Denote a = cos(0). First notice that Py = Py* and Pv2 = (1 I Now let us show that Pvv = v for any v Â£ V. If v Â£ V, it is represented in the form v = Pgu for some u Â£ U based on Lemma 2.52. But then Pvv = (l/a2)PgPuPgPgPjru = {l/a2)PgPjrPua2u PgP^u = v. We need to show now that Pytr1 = 0 for any v1 G V1. (2-51) Assume that PgPuPgv = v ^ 0. Evidently v V, and consequently, using Lemma 2.72, it can be represented as v = Pgu for some u E IA. Then we have Pg{PuPgV1 u) 0, which means that PuPgV1 u Â£ Qx- But we also have PuPgvL u = u' E U and we get u' G U D QL ^ {0}. It follows from the last equation that P^Pg\Tu' = 0 for v! G U. But this is a contradiction, since U is an eigenspace of the operator P?Pg |jf, corresponding to the eigenvalue cos2(0) f- 0. Consequently, we conclude that (2.51) holds. Now we investigate some properties of principal subspaces similar to those of principal vectors. Theorem 2.75 Let U C T and V C Q form a pair of the principal subspaces for subspaces T and Q corresponding to the angle 0 G 0(.F, Q). Then 6 Â£ 0p(P, Q) andU andV are the eigenspaces of the operators PjrPg\jr and PgPjr\g, respectively correspondng to the eigenvalue cos2 (9). 49 i Proof: Denote a = cos(0). Multiplying the first equality of (2.49) by Py from the right and the second equality by Pu from the left we get: (PfPv? = a 2PuPv, (PuPgf = c2PuPv. (2.52) From (2.52) we get PpPv PuPg, PgPu = PvPr- (2.53) Let us mention that the second equality of (2.53) is simply a conjugate of the first one. Now, multiplying the first equality of (2.49) by Pu and the second equality by Pv from the right we get: P?PvP?Pu ~ 02Pu, PgPuPgPv = &2Pv- (2-54) Using (2.53) in (2.54) gives P?Pg\fPu = v2Pu, PgPT\gPv = a2Pv, (2.55) which means that the assertion is proved. Theorem 2.76 Let 6 e Qp{P, G) \ {f} and U C T be an eigenspace of the operator PyrPg\jr, corresponding to the eigenvalue cos2(8). Then there exists a unique eigenspace V C Q of the operator PgP?\g, corresponding to the same 50 eigenvalue such that U. and V form a pair of principal subspaces corresponding to the angle 9. Proof: Consider a subspace V = PgU C Q. Based on Lemma 2.72 we conclude that V is an eigenspace of the operator PgP?\g, corresponding to the same eigenvalue. Next, using Lemma 2.74 we have the relation Pv = :T7o\PGPuPg- (2.56) cos2[6) To show that Pu jttw-Pf-PvPjt, cos2(9) we write: PjrPyPjr ---PjrPgP-pPU PjrPgPjr. COS Z{9) Using the fact that the operators P?PgPjr and Pu commute in the last equal- ity, we get: P?PvPf = P^PgPjrPu, which gives cos2(0)P^ = PyrPvPf. The last equality together with (2.56) show that U and V form a pair of principal subspaces corresponding to the angle 9. It remains to show that V is unique. If we assume that there exist two different subspaces Vi and V2 with the above mentioned properties, then we have from the second equality of (2.49): cos2{9)(Pv1 Pv2) 0- Since 9 < | we conclude that P^ = -Py2> which itself proves the assertion. It follows from Theorem 2.75 and Theorem 2.76 that Theorem 2.77 There exists a pair of principal subspaces for subspaces T and Q corresponding to a given angle 6 G 0(P, Â£?)\{f} if and only if9E p(P, Â£)\{f}. 51 Theorem 2.78 IfUcP and V C G form a pair of the principal subspaces for subspaces T and Q corresponding to the angle 9 G 0(P,G), then Q(U,V) ep(u,v) = {9}. Proof: Using Theorem 2.75 and equalities (2.53) we have PuPvPu = Pu(PjrPv)PjrPy = PuPqP?Pu = PuPrPgP.rPu = cos2(9)PU. This equality shows that U is an eigenspace of the operator PuPv\ui corresponding to the eigenvalue cos2(0), i.e. E(PwPy|w) = {cos2(0)}, which means that e(U,V) = {6}. (2.57) Next, using the same theorem and equalities, we have PvPuPv = Pv(PgPu)PgPv PvPjrPgPv = PvPgPjrPgPv = cos2(9)PV. This equality shows that V is an eigenspace of the operator PvPu\v, corresponding also to the eigenvalue cos2(8), i.e. E(Py^w|v) = {cos2(0)}, which means that (V,W) = {8}. (2.58) From (2.57), (2.58) we conclude that 0(1/, V) = 0p(l/, V) = {0}. Theorem 2.79 IfUcJ- and V C G form a pair of the principal subspaces for subspaces T and Q corresponding to the angle 9 G Ov(P, G) \ {f}, then U and V are isomorphic. Proof: If 9 = 0, then the assertion is trivial. If 8 G Ov(T, G) \ ({0} U {f }), using Theorem 2.43, we conclude that U and V are in generic position. But then, it follows from Theorem 1, [24] that U and V are isomorphic. 52 Theorem 2.80 If U(9),U() C T, and V(9), V(cf) C Q are the principal subspaces for subspaces J- and Q corresponding to the angles 9, {2-}, then the following relations hold: Pu{e)Pu() Puwnuw I Pv(6)Pv(4>) = Pv(6)nv() ! Pu(e) and Pv{4>) are mutually orthogonal if 6 ^ . Otherwise, if 6 = then we can choose V(8) such that Pu(e)Pv{0) = Pu(e)Pgl For given U{6) we can choose V(6) such that Pv(e)Pu(e) = Pv(e)Pr- Proof: Using Theorem 2.75 we conclude that the projectors considered in this theorem, are orthoprojectors onto the eigenspaces of the operators P^Pglr and PgPf Isj respectively. But then Pu{6) and Pu() commute, and so do Pv(e) and Pv{)- Next, let us mention that a product of two commuting projectors is a projector onto an intersection of their ranges, see Section 2.1. Consequently, the first two assertions are proved. To prove the third assertion, let us choose V(8) = PgU(6). Then we have Pu(6)Pv(6) = c"s2(0) Pu(9)PgPu(6)Pg = 'cos2('Â£) Pu(6)PrPgP^Pu(0)Pg = Pu(e)Pg- The proof of the fourth assertion is similar to that of the third assertion. Lemma 2.81 A pairU,V of principal subspaces corresponding to a given angle 9 ^ 0 forms a subspace U + V, invariant with respect to the orthogonal projectors P-p, Pg, I Pfr and IPg. Subspaces U (9)+V(9), corresponding to the different 9s, are mutually orthogonal. 53 Proof: The proof follows from Definition 2.71 and Theorem 2.80 Theorem 2.82 Let U and V be the principal subspaces for subspaces T and Q, corresponding to the angle 6 Qp(iF, Q) \ ({0} U {|}). Define U = X(PrPv\rx), V = X{PgPu\g). (2.59) Then U and V are closed and U_i, V are the principal subspaces for subspaces P~ and Q, corresponding to the angle \ 6; U, V_l are the principal subspaces for subspaces T and corresponding to the angle | 6; Vj_ are the principal subspaces for subspaces and C/, corresponding to the angle 9. Proof: We need to show that the ranges of the products of operators, given by (2.59) that is ^(Ppj.PvIjf-0 and *R(Pgj-Pu\g) are subspaces of TL. We use for this the third criterion of Theorem 2.2. Let us consider the first of two ranges given by (2.59). We have A = P?, B = Py> 91(^4) = P, and W(B) = V. Next, since 8 G Sp(P,G) \ ({0} U {|}), cos(0(ZY, V)) < 1. Using Lemma 2.32 and (2.3), we conclude that the third criterion of Theorem 2.2 holds. We have proved with this that 9K(Pjr Pv\jn.) is a subspace of Tt. The proof for 9\(PgPu\g) is completely similar. Next, evidently U . F1-, Vj_ Gx and 54 we have: PgP^PgPv = PgPV PgPTPgPV (1 COS2(9))PV = COS2(^ ~ 9)PV; PjrPgPjri.Pu = ------j-P:F-PgPjrxPvPTx = sin2(^)Pt-PqPt~PqPvP?x = PjrxPvPjrx = cos2( 9)PU\ PrPgxPfPu = PjrPu PtPqPtPu (1 cos2(9))PU = cos2(| 9)PU- PgxPfPgxPv = Y^CO^le)Pg PtPq-PuPg- = -^-^PgxPfPgxPjrPuPgx = PgPuPg = COS2( 9)PV\ Pf^Pg^Pp^Pux = (I ~ Pp-t-PgPr^Pux = cos2(#)P^x; Pg-LPjrA-PgPvx ~ {I ~ Pg^P^PgP}Pv = cos2(#)Pyx. Thus, PgPu b = cos2(| e)pv P^Pv\r = cos2(| d)pup PrPvJr = cos2(| 9)PU, PgxPu\gx = cos2(| 9)PV] PjrxPvx U-L = cos 2(9)PU, PgPu b-L = cos2(6>)Pv Let us investigate now a seeming difference between the formulae for the pairs of principal vectors for different pairs of subspaces (Theorem 2.62, equal- ities (2.45)) and pairs of principal subspaces (Theorem 2.82, equalities (2.59)). Namely, the question is do we obtain the same result, if we construct the differ- ent pairs of principal subspaces not based on equalities (2.59), but in a similar way to equalities (2.45)? The following theorem gives the positive answer to this question. 55 Theorem 2.83 Let it be the set of pairs of principal vectors {u, v} of subspaces F and Q, corresponding to an angle 6 Â£ 0p(P, Q) \ ({0} U {f}). Then U = span{v cos(0)u, {u, u} Â£ It}, Vj. = span{u cos(0)v, {, u} Â£ it}, (2.60) where U and V are defined by (2.59). Proof: Because of symmetry, we need to prove only the first of the two asserted equalities. As we have seen from Theorem 2.62, for a given pair {u, u} the vectors {u, w} (we can consider also two other pairs: {u, v } and {m_l, t>j_}), where = (v-cos(d)u)/sin(0), are the eigenvectors of the operators P^Pg 1.7^ and PgPp\g, respectively, corresponding to the eigenvalue sin2(0). Then from Lemma 2.72, we conclude that span{vcos(6)u, {u, r} Â£ 11} = P^V. Finally, using Lemma 2.74 we get Pu = P^-xPy Pjrx, which means that assertion is proved. Lemma 2.84 Let 9 G 0p(P, Q) \ {f} and Pu and Py be the orthogonal projec- tors on the principal subspaces U and V of the subspaces T and Q, respectively corresponding to the angle 6. Then the equality (Pjr Pg)\PU Py) = SHI\9){PU Py) holds. Proof: We have from Theorem 2.82: PfPgxPu = sin 2(6)PU, PgPf^Pv = sin 2(<9)Py. (2.61) Next, using equation (2.46), and the relations PjrPgPv 0 and PgPTPu 0, the assertion follows directly from (2.61). 56 Remark 2.85 Using Theorem 2.43, we conclude that if the subspaces T and Q are in generic position, then the assertions of Lemma 2.72, Lemma 2.74, Theo- rem 2.76, Theorem 2.77, Theorem 2.80, Theorem 2.82, Theorem 2.83, Lemma 2.84 hold without excluding the zero and right angles. 2.8 Principal Invariant Subspaces In this section, we generalize the definition of a pair of principal subspaces to a pair of principal invariant subspaces. Using a given pair of principal invariant subspaces of T, Q, we construct the pairs of principal invariant subspaces for different pairs of subspaces, such as 4F and Q1, TL and Q, 7F1- and QL. We investigate the question of uniqueness of the principal invariant subspaces. We also investigate the properties of isometries, given in Section 2.3. Namely, we show that these isometries map one subspace of a given pair of principal invariant subspaces onto another, and vice versa. We give also the relations between orthogonal projectors onto the principal invariant subspaces. Definition 2.86 A pair of subspaces U C T, VC Q is called a pair of prin- cipal invariant subspaces for subspaces T and Q, if the equalities PjrV = U, PgU = V. (2.62) hold. Now let us prove some auxiliary results. Lemma 2.87 Let U C P be an invariant subspace of the operator P?Pg \yr. Then V = PgU C Q is an invarinat subspace of the operator PgP?\g. Next, if 57 an operator PjrPg\u is invertible and its inverse is bounded, then the equality PyrPgljrU = U holds. If PgPp\v is invertible and its inverse is bounded, then also the equality PgPf \gV = V holds. Proof: Since U is an invariant subspace of the operator PpPg\j we have P?Pg \jU C U. From this relation we get PgPyr(PgU) C PgU, that is, V = PgU is an invariant subspace of the operator PgPjr\g. Next, since PfPg\u is invertible and the inverse operator is bounded, we conclude that PjrPg\pU = U. Similarly, if PgPjr|v is invertible and its inverse is bounded, then we have PgPp\gV = V. The following lemma gives a generalization of the relation (2.50) for invariant subspaces. Lemma 2.88 Let U be an invariant subspace of the operator PjrPgljr and Pu be an orthogonal projector on this subspace. Further assume that the operator PpPg\u is invertible and its inverse is bounded. Then the orthogonal projector Pv on the invariant subspace V = PgU of the operator PgPy?\g is given by Pv = PgPu{P?Pg\u)~l PuPg- (2.63) Proof: First notice that Py = Py* and Pv2 = PgPu{PrPgPr)-lPuPgPgPu{PTPgPr)-lPuPg = Pg Pu ( Pf Pg P?)~1 Pf Pg P? Pu2 (Pf Pg Pt )~ ^1 PuPg = PgPu(PrPgPr)-XPuPg = Pv 58 Further, if v G V then it is represented in the form v = Pgu, where u Â£U. We have: Pw = PgPu{P^PgPT)~lPuPj{PgP?u) = PgPuiP^PgPj^y1 PrPgPrPu'u = PgPuu = Pgu = v. We need to show now that Pyt~ = 0 for any d1 G V1. Assume that PgPu(PrPg\u)~1PuPgv v ^ 0. Evidently v G V, and consequently, it can be represented as v = Pgu for some u G ZP Then we have Pg(Pu(PjrP>g\u)~1PuPgv u) = 0, which means that PuiP^PgluY1 PuPgv1- uG Q1- But we also have Pu(P^Pg\uYl PuPgv-1 u = u' eU and we get u EU 4- {0}. But this is a contradiction, since the inverse of the operator P?Pg\u is not bounded. Lemma 2.89 If U C T and V C Q form a pair of the principal invariant subspaces for subspaces IF, Q and | ^ 0(77, V), then the operators PrPg\u, PgP?|v ore invertible and their inverses are bounded. Proof: We have 0 ^ H(PuPv\u) and 0 ^ Â£(PyP^|v)- Using the same argument is in the proof of Theorem 2.33, we conclude that Â£(PwPy|w) = Â£(PyPw|y) is a compact subset of [0,1], separated from 0. This means that there exists an m > 0 such that ||PjrPg|jFu|| > m||u|| for every u G T and ||PgPjr|er|| > m||u|| for every v G Q. Finally, using Corollary 3 ([77], p. 43), which states that an operator T G B(H) admits a continuous inverse T-1 if and only if there exists a positive constant 7 such that ||Tu|| > 7||u|| for every u from the domain of T we conclude that the assertion is proved. The following lemma provides other relations between the invariant and principal invariant subspaces. 59 Lemma 2.90 Let U C P be an invariant subspace of the operator P? Pq\f> V = PgU, and | ^ Q(U,V). Then U and V form a pair of principal invariant subspaces for subspaces P and Q. For the given subspace U, the corresponding subspace V is unique. Proof: From Lemma 2.89 we conclude that the operators PfPg\u, PgP?\v are invertible and their inverses are bounded. Then, using Lemma 2.87, we have PtPq \jrlA = U, which means P?V U. Similarly, we get PgU = V. Finally, we conclude that U and V form a pair of principal invariant subspaces. The uniqueness of V follows from the equality V = {PjrPg\u)~lU. m Now we investigate some properties of principal invariant subspaces similar to those of principal subspaces. Theorem 2.91 IfUcP and V C Q form a pair of the principal invariant sub spaces for subspaces P and Q, then U and V are invariant subspaces of the operators PrPg\r and PgP?\g, respectively. Proof: We have using Definition 2.86: PTPg\TU = Pr{PgU) = P?V = W; PgPAeV = Pq(PfV) = PgU = V. Theorem 2.92 Let U and V be the principal invariant subspaces for subspaces P and Q and 0, | ^ @(Â£Y, V). Define U = m(P^PV \px), Vj. = K(PgPU\g). (2.64) 60 Then h(_i, V are the principal invariant subspaces for subspaces F~ and Q; hi, Vj_ are the principal invariant subspaces for subspaces F and Q1; Uj_, Vjl are the principal invariant subspaces for subspaces F1- and Q1. Proof: We have max{cos(0(W, V))} < 1, since by the assumptions 0, | ^ 0(ld, V). Consequently, using Lemma 2.32 and (2.3), we conclude that the third criterion of Theorem 2.2 holds. The remaining part of a proof is similar to that of Theorem 2.82. The following assertions connect the sets of angles between a given pair of subspaces and a pair of their principal invariant subspaces. Lemma 2.93 If Id C IF and V C Q form a pair of principal invariant subspaces for subspaces F and Q, then e{u,v)ce{F,g). Proof: Using Theorem 2.91, we conclude that U is an invariant subspace of the operator PjrPg\jr and V is an invariant subspace of the operator PgP?\g. Then we have the decompositions PfPgW = PuPv\u PjreuPgev\r&u and PqPt\g = Pv^lv PqqvPt&uleev- 61 From these equalities, using Theorem 7.28, p. 208, [73], mentioned in Section 2.4, we get ^{PfPgW = Â£(PuPv\u) U S(PjrewP6ev|jrw) and S(PgPr\g) = S(PyP^lv) u S(PgevPjre^|gev), that imply 0(7/,V) C 0(P,Q) and 0(V,IA) C 0(C?,V). These two inclusions prove the assertion. Lemma 2.94 Let U C T and V C Q be a pair of principal invariant subspaces for subspaces J- and Q, and U_cU,V_cV be a pair of the principal invari- ant subspaces for subspaces U and V. Then U_, V form a pair of the principal invariant subspaces for sub spaces TQ, and e(ZY,v) c (w,v) c (p,g). Proof: The proof follows directly from Definition 2.86 and Lemma 2.93. Now, let us investigate the relationships between principal invariant sub- spaces provided by (partial) isometries from Section 2.3. In the following theo- rem we use the fact of existence of a polar decomposition which is guaranteed by Lemma 2.15. Theorem 2.95 Let {U, V} be a pair of principal invariant subspaces of the subspaces T, Q, and Pu, Py be the orthogonal projectors on these subspaces, respectively. Assume also that PgPf = WPfPgPf (2.65) 62 is the polar decomposition of the operator PgP?, with W (partial) isometry. Then: 1. V = WU; U = W*V; 2. Pv = WPUW*; Pu = W*PVW. Proof: We prove only the first parts of the both assertions; the second parts are analogous. It follows directly from Definition 2.86 that PfPgPfU = IA (see also The- orem 2.91). From this equality we obtain that \J PjPgPjAA = U. Multiplying both sides of (2.65) by U and using Definition 2.86 again, we have: V = PglA = PgPjAA = Wy/PrPgPjJA = WU. From the equality PgPu PyPr we have PgP?Pu = P^PgPjr. Next, using (2.65), we have WsJPTPgPTPU = PvWy/PTPgPT. (2.66) Using Theorem 2.91 again, U is an invariant subspace for ^/PyrPgPjr, and con- sequently 'JPrPgfrPu Puy/PfPgPr- Then we have from (2.66) WPU = PvW, multiplying both sides of which by W* from the right and using the fact that WW* is a projector onto the subspace of Â£/, which contains V as a subspace, leads to the first part of the second assertion. Remark 2.96 Using Theorem 2.16, we conclude that if the inequality (2.13) holds, then the assertions of Theorem 2.95 hold for both choices ofW = W and W = W, where W and W are defined by (2.14) and (2.17), respectively. 63 2.9 Principal Spectral Decompositions Before we investigate principal spectral decompositions of interesting oper- ators, we formulate the definitions and basic properties of the spectral family and a spectral measure corresponding to a given selfadjoint operator. Next, we give Lemma 2.88 in terms of a principal spectral decomposition. We also show how the isometries given in Section 2.3 link the spectral families of two products of projectors. Definition 2.97 ([37, 62]). A one-parameter family of orthogonal projectors depending on a real parameter X with properties a) E(A) < E(p), or equivalently, E(X)E(p) = E(A) for A < p; b) E(X + 0) = E(X); c) E(A) = 0, A < m and E(A) = I, X > M for some given pair of real numbers m < M is called a spectral family. Definition 2.98 (Definition C.10, p. 380, [10]). Let X be a compact subset of real line. The set of the smallest o-algebra of subsets of X that contains all open sets is called a Borel set. Theorem 2.99 ([62]). Every selfadjoint transformation T in Hilbert space, with greatest lower and least upper bounds of E(T) equal to m and M, uniquely 64 defines a spectral family such that pM T= XE(dX) (2.67) d m0 holds. Let us now describe in a nutshell a method of constructing a spectral measure from a spectral family, e.g., [37]. First, for any semiclosed interval J = (A', A"] we set E(J) = E(A") E(A'), which is the orthogonal projector onto the subspace fK(Â£(A")) A')). Evidently, E{J')E{J") = E(J")E{J') = 0 for disjoint For any real A, we set P{A) = Â£(A) E(A 0), which is a projector onto the subspace 9^{E(A)) ^(^(A 0)). If 8 is the union of a finite number of intervals (open, closed or semiclosed) on the real line, 5 can be expressed as the union of disjoint semiclosed intervals and points {A}. Let us define E(S) as the sum of the corresponding E(J) or P(A). Then E(8) is called a spectral measure on the class of all sets 8 described above. This measure E(5) can then be extended to the class of all Borel sets of the real line by a standard measure-theoretic construction, e.g., [16]. The properties of the resulting spectral measure are given by the following relations, where 8, 8' and S'1' are arbitrary Borel sets: 1. E(8' n8") = E{8')E(8")-, 2. E{8' U 8") = E{8') + E(8") E(8')E(8"); 3. E(E(T)) = /; 4. E(0) = 0; 65 5. E{S)T = TE(S); 6. E(Ts) C 5, where T = T\e^)h is the restriction of T to the subspace Ti = E(5)Tt, and 8 denotes the closure of 5; 7. E*i E{Si)x = ^U^jx, Vx H. The first four properties are the general properties of orthogonal projectors. The last three properties determine the corresponding to T spectral measure. We can summarize the above construction of a spectral measure by the following assertions: Lemma 2.100 ([16], p. 897, Corollary 4)- A bounded selfadjoint operator T uniquely determines a countably additive spectral measure E on the Borel sets of the real line which vanishes on the resolvent set of T and has the property that /(T)= [ f(X)E(d$$, feC(S(T)h
Je(t)
where C(H(T)) is the space of continuous functions on E(T).
Definition 2.101 ([16]). The uniquely defined spectral measure, associated in
Lemma 2.100 with the selfadjoint operator T, is called the resolution of the
identity for T.
Lemma 2.102 ([16], p. 898, Corollary 7). Let E be a countably additive spec-
tral measure defined on the Borel sets of the real line. Then E is the resolution
of the identity for the bounded selfadjoint operator T if and only if the equality
T= f XE(dX).
XÂ£(T)
holds.
66

Lemma 2.103 ([10]). If T = XE(dX) is a spectral decomposition of a
selfadjoint operator T, then E(5)TC is a invariant subspace for every Borel set
6.
Using Lemma 2.103, we can introduce following
Definition 2.104 Let C Q(P, Q) be a Borel set, and {Â£1}, {E2} be the
spectral measures corresponding to the operators P^PgPjr and PgPjrPg, respec-
tively. Then U(0) Ei(cos(Q))H and V(0) = E2(cos(Q))H are called a pair
of principal invariant subspaces corresponding to the set of angles 0.
Now let 0 C Q(P, Q) be a Borel set and denote by Pu(e) and PV(e) the or-
thogonal projectors onto the invariant subspaces U(Q) and V(0) corresponding
to 0.
The following lemma and theorem give a generalization of the results of
Lemma 2.54 and Lemma 2.60.
Lemma 2.105 Let 0', 0" C (Â£, ^)\{f} be the Borel sets. Then the following
equality
Pu(e')Pu(Q") = Pu(e'ne") (2.68)
holds.
Proof: Denote by {Â£} a spectral measure of the operator PjrPgPp. Then,
using Lemma 2.103 we have U(&) = E(cos(Q'))H and U{Q") = Â£(cos(0"))7L.
But then, using the first property of spectral measure, we get:
Pu(e')Pu(&') = Â£(cos(,))Â£(cos(0")) = Â£(cos(0') fl cos(")) = Pu{e>ne"),
which is (2.68).
67

Theorem 2.106 Under assumptions of Lemma 2.105 the following relation
Pu{e')Pv(e") Pu(er\')Pg
(2.69)
holds.
Proof: Based on Lemma 2.88 and on Lemma 2.105 we have:
Pu(e')Pv(e) Pu(e')PgPu{e"){PrPg\u(e")) Pu(B")Pg =
Pu(e'){PjTPg\u(&'))Pu(Q"){PTPg\u(B)) 1Pu{e")Pg =
Pu{B)Pu{e")Pg = Pu(Q'ne")Pg-
Corollary 2.107 If for Borel sets ' and 0", 0' n 0" = 0 then
Pu(e')Pv(e") = 0,
that is, we have byorthogonal projectors.
The following theorem gives the result of Lemma 2.88 in terms of a principal
spectral decomposition.
Theorem 2.108 Let {Pi} and {E2} be the spectral measures of the operators
PjrPg\jr and PgPf \g, respectively. Further, let C 0(P,Q) \ {f} be the Borel
set, Pw(e) = Jcos{e) Ei(dX) and PV(e) = /cos(e) Pi{d\). Then the following rela-
tion
fv(e) = { / )eiMlPs (2.70)
Ucos(e) A J
holds.
68

Proof: We have using Lemma 2.100
[ Fi(dA) {Pj^Pg\u{e)} 1 Pu(&){PfPg\u(e)} lPu(e)-
Jcos(0) ^
Based on this equality and on Lemma 2.88 we conclude that equality (2.70)
Now we obtain a result which generalizes the relation between orthogonal
projectors onto the principal invariant subspaces of subspaces T, Q obtained in
Theorem 2.95.
Theorem 2.109 Let E\{s) and E2(s) be the spectral families of the operators
PfPgPf and PgPpPg, respectively, and the polar decomposition of the operator
PgPjr is given by (2.65). Then the relationship between the spectral families
E\(s) and E2(s) is given by
Proof: Denote F(s) = W Ex(s)W*. We need to show that (F(s), s R} is a
spectral family and that
To show that {F(s), s E R} is a spectral family, we need to prove the following:
F(s) is an orthogonal projector for any s R. We have: (F(s))* =
F(s) and (F(s))2 = WEi{s)W*WEi(s)W*. The product W*W in the
last expression is equal to P? or to the identity. In both cases we have
(F(s))2 = W(F!(s))2W* = WEx{s)W* = F(s)]
holds.
E2{s) = WEx{s)W*.
(2.71)
(2.72)
69

F(s)F(t) F(s) if s WEx(s)W* = F(s).
F(s + 0) = F(s) for s Â£ i?; this is evident, because E\(s + 0) = Ei(s);
lim^-co F(s) = 0 and lim^oo F(s) = I. These relations are also immedi-
ate consequences of the relations linrs_>_oc E\(s) = 0 and lims_>00Â£1(5) =
I.
It remains to show that (2.72) is true. For W = W we have:
[ sF(ds) = W f sEl(ds)W* = WPTPgPTW* =
Jr Jr
Wy/PrPgPriWy/PrPgPrY = PgPT{PgPTy = PgPfPg. (2.73)
Remark 2.110 Using Theorem 2.16, we conclude that if inequality (2.13)
holds, the assertion of Theorem 2.109 holds for both choices of W = W and
W = W, where W and W are defined by (2.14) and (2.17), respectively.
70

3. Discrete Angles Between Subspaces
In this chapter, we formulate the definition and basic properties of s-numbers
of bounded operators on Hilbert space following [22] and define the discrete an-
gles between subspaces using s-numbers. Similar to the general case of Chapter
2, we introduce principal angles, principal (invariant) subspaces, and investigate
their properties for discrete angles. Discrete angles are the subset of angles of
Chapter 2, characterized by variational principles.
3.1 Definition and Basic Properties of s-Numbers
In this section, we remind the reader of the definition of s-numbers, and
give their basic properties, following [22]. We show that s-numbers can be
equivalently obtained by Courant-Fisher max-min principle. We also provide
some relations between the s-numbers of different operators that we need later.
Let H be a Hilbert space and A B(H) be a bounded linear operator
defined on H. Let us present here the known definitions and results related to
s-numbers, e.g., [22, 61].
Definition 3.1 [22]. A point of the spectrum of self adjoint operator T B(H)
is called a point of the condensed spectrum, if it is either an accumulation point
of the spectrum of T or an eigenvalue of T of infinite multiplicity.
Remark 3.2 The above definition is the same as definition of an essential spec-
trum of self adjoint operator, see e.g., [73], p. 202. Consequently, we denote this
subset o/E(T) by Ee(T).
71

Denote by /x the supremum of the spectrum of T Â£ B(H). If the point p,
belongs to Ee(T), we set
3 = 1,2,....
If the point /x does not belong to Ee(T), then it is an eigenvalue of finite multi-
plicity and in this case we set
Aj(T) = V, j 1) 2,.. ,p,
where p is the multiplicity of the eigenvalue /x. In the latter case the subsequent
numbers Aj(T), j = p + 1,... are defined by
^p+j{T) Aj(Ti), j = 1,2,...,
where the operator 7\ is given by
T\ = T iiP,
and P is the orthogonal projector onto the eigenspace of the operator T corre-
sponding to the eigenvalue p.
Definition 3.3 The s-numbers of an operator A Â£ B{Ti) are defined by
Sj(A) = \j{>/aFA), j = 1,2,..., (3.1)
and Xj(-) are obtained by the above described procedure with T = \JA*A.
According to Definition 3.3, the sequence of s-numbers {sj(A)} of an ar-
bitrary operator A Â£ B(7i) is nonincreasing and
lim Sj(A) = Soo(^), (3.2)
j-*oO
72

where Soo(A) denotes the number A^T) = sup{A | A G Se(T)} (see [22], p.
60).
Also, from the definition of s-numbers it follows (see [22], p. 61) that for
any operator A e B(TC)
Sj(A) = Sj(A*), j = 1,2,... (3.3)
and for any scalar c
Sj(cA) = |c|s.,-(A), J 1,2, (3.4)
The following lemma shows that the above described procedure for obtaining
Aj{T), j = 1,2,... gives us the Courant-Fischer values, e.g., [31, 36, 39, 61], of
an operator T.
Lemma 3.4 Let U be an eigenspace of a selfadjoint operator T G B(H) corre-
sponding to an eigenvalue A, and Pu be the orthogonal projector onto U. Then
the following equalities
T XPu = (I PU)T{I Pu) = (I- Pu)T = T(I Pu)
hold.
Proof: Since T = T* and U is an eigenspace of T, then U is an invariant
subspace, e.g., [10] for the operator T and we have
TPU = PuT PuTPu = XPu- (3.5)
But then T XPu = T(I Pu) and the assertion follows from the fact that T
commutes also with I Pu-
73

Let us now consider the Courant-Fischer principle and the properties of the
corresponding extremal values for bounded operators. First, we consider the
Courant-Fischer values for a bounded selfadjoint operator A from the top:
Aj[(T) = sup inf (u,Tu), k = 1,2,.... (3.6)
nkcn ,uWfc
dimHk=k llll = l
The resulting sequence A*(T) is ordered nonincreasingly:
A}(r)>A(T)>....
Second, we consider the Courant-Fischer values from the bottom:
A[(r) = inf sup (u,Tu), k = 1,2,.... (3.7)
,7ikSHu u^nk
dim -Hk=k ||w||=1
The resulting sequence A^(T) is ordered nondecreasingly:
a!(t) It is known, e.g., [61], Theorem XIII.1, p. 76, that for the Courant-Fischer
values from the top (bottom) there exist only two possibilities. For each fixed k
either:
there are k eigenvalues (counting the multiplicities) above(below) the
top(bottom) of the essential spectrum, and AÂ£(T) (Aj.(T)) is the A:th eigen-
value counting multiplicity;
Aj.(T) (A[(T)) is the top(bottom) of the essential spectrum, that is Aj.(T) =
sup{A | A Â£e(T)} (A\{T) = inf{A | A e Ee(T)}) and in that case
Al(T) = Aj.+1(T) = ... (A[(T) = Xl+1{T) = ...) and there are at most
k 1 eigenvalues (counting multiplicity) above(below) AÂ£(T) (Aj.(T)).
74

Lemma 3.5 Let T G B(H) be selfadjoint, and a be a real number. Then the
following equality
Aim = a \i(al ~ T) (3.8)
holds.
Proof: We have:
a Xlk(al T) = a sup inf (u, (al T)u) =
nkcn ^
dimllull1
a sup inf [a (u,Tu)\ = a sup [a sup (u, Tu)\
nkcn nkcH uenk
dimTik=k l!ull1 dim7i:/c=A: ||u||=l
inf sup (u,Tu) = Aj(T).
d\mHk=k ||u||=1
Corollary 3.6 Let T G B(H) be selfadjoint, and a be a real number. Then the
following equality
Aim = a Ai(ai T) (3.9)
holds.
Corollary 3.7 IfT G B(H) is selfadjoint, then
Aim =-Ai(-n = 1,2.......... (3.10)
The following lemma gives a relation between the Courant-Fischer values of
two interesting operators.
Lemma 3.8 For any k = 1,2,... the following equality
Ai(iVVM = l XiiPrPM (3.11)
holds.
75

Proof: The proof of this lemma follows from Corollary 3.6 with a = 1,
T = P?Pg\r and PjPg\? + Py:Pg\?r = Pf\? = Iy.
3.2 Discrete Angles Between Subspaces
Here we give two equivalent definitions of discrete angles between subspaces.
The first definition is recursive, and the second one is based on Courant-Fischer
numbers of some relevant operators. We consider sequences of discrete angles
between different pairs of subspaces.
Let P and Q be proper nontrivial subspaces of H, and Py and Pg be the
orthogonal projectors onto P and G, respectively.
Definition 3.9 Denote q = min{dimP, dim G}. The smallest discrete principal
angles 6i,...,6q Â£ [0,tt/2] between P and G are defined recursively for k =
by
cos (8k) = supsup|(u,u)| = \{uk,vk)\ (3.12)
uef veg
subject to
IMI = Illll = 1, (u,Ui) = (v,Vi) = 0, i = 1,..., k 1,
provided that the pair of vectors Uk,Vk exists. If for some k = n the pair Uk, Vk
does not exist, we set 9k = 9n for all k > n.
If q = oo, we have a sequence of discrete angles 9\, 62,---
Theorem 3.10 Let P and G be the subspaces of a Hilbert space H, P?, Pg be the
orthogonal projectors onto these subspaces, respectively, and 8k, k = 1,2, ...,q
be defined by (3.12). Then $k = arccos(sk(P^Pg)), k = 1,2,..., q. 76 Proof: First let us prove that cos(6>i) = Si{Pj:Pg) = II PjrPg sup sup |(u,u)|. uaT vaQ Ml=i IN=i (3.13) We use for the proof a bilinear form representation of an operator (see, e.g., [73], Theorem 5.35, p. 120). Then we have: \\PfPq\\ = sup sup \{u,PfPgv)\ = sup SUp \(PjrU, Pgv)\ = uan van uan van ||u|| = l |M| = 1 ||u||=l IMI=1 sup sup \(Pjru, Pgv)\ = sup sup |(u,n)|. uaf vaQ uaf vaQ ||u|| = l |M| = 1 IMI=i Nl=i Next, using Lemma 3.4 we get: COs{6k) = Sk(PTPg) sup uan ||u||=l uUk_- IK \PgPr\u)\ SUp |K (/ Py^PgPjr^I PWjfc_| = uan \\u\\=l sup I {{I ~ Pv^Ju, PgPr{I ~ Puk_x)u)\ = uan IMI=i sup \{Pg{I Pvk-x)u, PA1 Puk_iKI = uan ||u||=l SUp \{PgQVk_1U,P^euk_lU)\ = uan IMI=i sup sup \(u,v)\ = (uk,vk) = cos(6k). uaF vaQ |]u||=l |M| = 1 uUk-1 Since the s-numbers are defined by (3.1), using the assertion of Theorem 3.10, we can introduce the following definition, equivalent to Definition 3.9. Definition 3.11 The sequence el(P,G) = {dk: 6k = aiccoa(y/\lk(PrPg\r)), k=l,2,...,q} (3.14) 77 where Xl(PfPg\?) are the Courant-Fischer numbers of the operator PtPq I^j given by (3.6), is called the sequence of smallest discrete principal angles between sub spaces T and Q. To define the sequence of largest discrete angles we need the intersec- tion of two sequences, which we define in the following way. Assume that we have two sequences of real numbers, one of which contains in the begin- ning a finite number of terms that do not belong to other sequence. Remain- ing parts are the same. If we denote the sequences by and then there exists minimal positive integer k such that {rq,... rk,ti,t2,... } = {r*i, 7~2, }, or {ti,.. .tk,ri,r2,...} = {ti,t2,..If there exists such k for the first equality, then we call 1 D Otherwise we set {n}t=i = {*i}Â£i nfo}^. Definition 3.12 The sequence Qld(F,G) = {#* : 0fc = arcsin(v/^^i), k = l,...,dim^r| nj#* : 0k = arcsin X^PgP^ A: = 1,2,... ,dimÂ£/j (3.15) where X^PpPgx^) and Xjc(PgP:F\g)are the Courant-Fischer numbers from the top of the operators P^Pg\jr and PgP;r\g, respectively, given by (3.6), is called the sequence of largest discrete principal angles between subspaces T and Q. It follows from Definition 3.11 that Q\(!F,G) = Qd(G, T) and this fact is discussed in detail in Theorem 3.14. If we defined the largest discrete angles using only one product of projectors, the result would not be symmetric with respect to the subspaces. To make it symmetric, we consider the intersection of the sequences. 78 Remark 3.13 Let us consider the sequence of discrete angles as the set of its components. Then we have the following relationships between these sets and the sets of angles defined in general case by Definition 2.19: Qd{lF, G) C G), eld(?,g) c e(r,g). ifq < oo, then erd(r,g) = eld(^,G). The following theorem provides a detailed description of the sequences of discrete angles. For readers convenience, we repeat here the notation of sub- spaces, that are used in decompositions of the space, (2.4), (2.5), (2.6), (2.8). Theorem 3.14 Let T and Q be subspaces ofTL, 9Jtoo = T C\ Q, 9JT0i = T fl G, m10 = n g, Tiu = n g1, 3n = m0 mu m0 = F (OToo 9Jt01), 97li = TL 0 (SDTio 0 O^n), Wl'0 GO (9Jtoo 0 9Hio)- Denote for transparency of the correspondence by$ = SDlo and 0 = Then:
1.
e^,0)-(o.........o,e'(3,0), |.......Â§), (3.16)
where number of Os is equal to dimSDloo, number of angles in ^(S,0) is
equal to dim S' = dim<5, number of is equal to
min{dim 9Jt0x, dim 3Jt10}.
If any of SJloo or 5 are infinite dimensional, then the next part does not
appear in the sequence.
2. e'jg.T) = e'd(?,g);
3.
el(r, S) = (|, ,ej(5, e). o....o). (3.17)
79

where number of ^s is equal to
min{dim QJloi, dim OJtio},
number of Os is equal to dim OToo- If any of previous sub spaces is infinite
dimensional, then the next part does not appear in the sequence.
i i(.s,r) = e>d(f,g);
5.
ei(^, ex) = (o,..., o, | e(s, e), ..., (3.is)
where number of Os is equal to dim9Jtoi, number of |s is equal to
min{dim 9EFt0(b dim 9Jtn}.
If any of previous subspaces is infinite dimensional, then the next part does
not appear in the sequence.
6.
eiP7. Sx) = (f, -. f. | e}(5, e), o,, o), (3.i9)
where number of |s is equal to
min{dim9ftoo, dim 9Jtn}>
number of Os is equal to dim Q7l0i If any of previous subspaces is infinite
dimensional, then the next part does not appear in the sequence.
7.
i(^x.ex) = (o,... ,o, ej(3.e), . .(3.20)
80

where number of Os is equal to dim97tu, number of^s is equal to
min{dim Wlw, dim 9Jt0i}
If any of previous subspaces is infinite dimensional, then the next part does
not appear in the sequence.
8.
ej(^x, e1) = (|,..., ei(s, e), o,..., o), (3.21)
where number of |s is equal to
min{dim fOlio, dim 9Jl0i},
number of Os is equal to dimfUtn. If any of previous subspaces is infinite
dimensional, then the neoct part does not appear in the sequence.
Proof: Let us prove only the first assertion. The proof of other assertions is
similar. Following Defininition 3.11, we need to consider the Courant-Fischer
numbers of the operator PjrPg\j? from the top. Let us consider the decomposi-
tions (2.10) not only for P?Pg\r, but also for PgP?\g. We have:
P^Pg\x = P^Pglmso PrPg\xrkn PfPg\m = -f| PgPAx PgPrlanoo PgPAwho PgPAxn = ^Iotoo 0|otiO PrPg\sai-
The first parts I\moa gh-e the cosines equal to 1, that is the zero angles and
their number is equal to dim 2Jl0o- The second parts 0|$oto1 and 01 auio 5 give the cosines equal to 0, that is the right angles and their number is equal to min{dim3Jl0i, dimSUlio}. The third parts both give the angles ), and 81 their number is equal to dim# = dim0. If we consider only one decomposition, that is if we follow to Definition 3.11, then we consider only k = 1,2 angles, which results in the same sequence of the angles. Because of this we do not need to consider the intersection of the sequences for the smallest angles. 3.3 Principal Vectors, Principal Subspaces and Principal Invariant Subspaces Corresponding to the Discrete Angles In this section, we define, similar to the general case of Chapter 2, the prin- cipal vectors, principal (invariant) subspaces and discuss their properties. The subset of discrete angles is determined, for which all properties of the princi- pal vectors and principal subspaces from the general case also hold for discrete angles. The set of principal invariant subspaces in case of discrete angles is described. The definitions of principal vectors and principal subspaces are similar to Definition 2.50 and Definition 2.71. Definition 3.15 Normalized vectors u = u(9) G P and v = v(6) Q form a pair of principal vectors for subspaces T and Q corresponding to the angle 6 e &d(P,G), if the equalities Pjtv = cos(0)it, Pgu cos (6)v (3.22) hold. Definition 3.16 A pair of subspaces U C F, VC Q is called a pair of prin- cipal subspaces for subspaces T and Q corresponding to the angle 9 Â£ Qd(P, Q), if the equalities P?Pv\r = cos2{9)Pu, PgPu\g = cos2(0)Pv (3.23) 82 hold. In Chapter 2 we have shown in Theorem 2.57 that there exists a pair of principal vectors for subspaces T and Q corresponding to a given angle 6 Â£ Q(-T,G) \ {f} if and only if 8 Â£ QP(F,G) \ {f}; and in Theorem 2.77 that there exists a pair of principal subspaces for subspaces T and Q corresponding to a given angle 6 Â£ G(T,G) \ {|} if and only if 6 Â£ QP(^F,G) \ {Â§} Let us consider the set dp(^, G) = {i(^, G) u i(.F, G)} n Qp(f, G) (3.24) of all discrete angles, cosines of which also are the eigenvalues of the correspond- ing operators. Then it follows directly from Theorem 2.57 that Theorem 3.17 There exists a pair of principal vectors for subspaces T and Q corresponding to a given angle 6 Â£ d(^, G) \ {Â§} if and only if 6 Â£ ^(.F, G) \ {Â§} It follows directly from Theorem 2.77 that Theorem 3.18 There exists a pair of principal subspaces for sub spaces T and G corresponding to a given angle 6 Â£ Qd(^, G) \ {f} if and only if 6 Â£ Qdpi^i G) \ {f} For the angles belonging to QdpifT', G) defined by (3.24), all properties of principal vectors and principal subspaces given in Section 2.6 and Section 2.7, hold. Let us consider now the principal invariant subspaces in the discrete case. We need to mention here the following: Qdpi^T, G) can be empty, finite, or 83 countably infinite. Definition 3.19 A pair of subspaces U C T, V C Q is called a pair of principal invariant subspaces for subspaces T and Q, if at least one of them is spanned by an arbitrary subset of the discrete principal vectors of the corresponding subspace, corresponding to a subset ofQdpi^, Q) \ {f} and the equalities PTV = U, PgU = V (3.25) hold. Theorem 3.20 Let U C T be spanned by the set of discrete principal vectors {tij}, 1 < j < J, where J < oo, and uj corresponds to a discrete angle 8j Â£ 0dp(^\Â£7)\{f}- Then the subspace V C Q which withU forms a pair of principal invariant subspaces, is spanned by {vj}, 1 < j < J, where {uj, Vj} form a pair of principal vectors, corresponding to the angles 8j, 1 < j < J. Proof: Consider a set of vectors {vj} with Vj = (1/cos(6j))PgUj, 1 < j < J. Using Theorem 2.56, {uj,Vj} form a pair of (discrete) principal vectors, corresponding to 6j, 1 < j < J. For arbitrary v E V, based on Definition 3.19, there exists u E U such that v Pgu. If u = i juji then v J2j=i Â£j'vji where = Â£j cos(8j). 84 4. Estimates for Angles Between Subspaces In this chapter, we prove an estimate for Hausdorff distance between the spectra of absolute values of bounded linear operators. This estimate can be used for getting other estimates for the proximity s-numbers of bounded operators. Next, we obtain the estimates for the proximity of squared cosines of the angles. 4.1 Perturbation Bound for the Spectrum of an Absolute Value of a linear Operator As we have seen in Section 3.1, the s-numbers of an operator are defined using the absolute value of the operator. Next, based on s-numbers, we defined the discrete angles in Section 3.2. On the other hand, there exists an estimate for the distance between the spectra of two selfadjoint operators, which is given below. Consequently, having the estimate of the distance between the spectra of the absolute values of the relevant operators, is one possible way of estimating the proximity of the sets of angles between subspaces. We obtain an estimate of this kind here. By dist(Si,S2) we denote a Hausdorff distance, e.g., [37] between the sets Si and S2 on a real line, which is defined by dist(Si, S2) = max{sup d(u, S2), sup d(v, -Si)}, uS2 and d(u, S) = infes \u v\ is the distance between a point u and a set S. Let A,B B{H) be the linear bounded operators. It is known (see, e.g., [37], [59]) that if A and B are selfadjoint operators, then the following inequality 85 holds: dist{E{A),E{B))<\\A-B\\. (4.1) The aim of the next theorem is to get a bound on the spectral perturbation for an absolute value of a linear bounded operator. Theorem 4.1 Assume A and B are linear bounded operators on a Hilbert space 7i and the equalities J2(AA*) = Â£(.AM) and Y,(BB*) = T,(B*B) hold. Then the following inequality holds: *s(S(|A|).E(|B|))<||/l-i?||. (4.2) Proof: Consider an operator A = 0 A* 1 V (4.3) acting on Tt 0 Tt. First let us show that the spectrum of the operator A is a symmetric subset of the real axis, that is if a E(T) then also o Â£ E(j4). That E(A) is real follows from the fact that A is selfadjoint. Next, let us assume that o Â£ P(A), where P{A) is the resolvent set and show that a Â£ P(A). If a Â£ P(A) then there exists (A alnen)-1- Denote this inverse operator by -E D Direct multiplication shows that then / V E D C -F \ the operator C F \ 86 A2 = \ is the inverse to the operator (A + alnn)- Direct multiplication also shows that / AA* 0 0 A* A or A2 AA* A*A, which is the same. This equality gives E(A2) = E(AA*) U E(AM) = E{AA*) = Z(A*A). Since |^4j2 = AA* we conclude that if a G E(|A|), 0, then a2 e T,(AA*) and vice versa. On the other hand we have E(j4) = E(j4|) U {E(|.A|)}. Next, based on similar conclusions for B and the fact that the operators A and B = 0 b'' \B , are selfadjoint, from (4.1) we conclude that the inequality 7 disi(E(i),E(Â£)) < ||i-B| (4.4) holds. It remains to mention that symmetricity of the sets E(A) and E(B) gives dist(T,(A),E(B)) = dist(E(|A|), E(|B|)) and that ||i 5|| = \\A Â£||. Remark 4.2 The operator (4.3) is widely used. According to [70] its present popularity is due to Wielandi (see [18], p. 113) and Lanczos [45]. 4.2 Estimate for Proximity of The Angles In this section we give an estimate for the proximity of squared cosines of angles from one subspace to another. The following theorem gives us an estimate for the proximity of the set of squares of cosines of >(JF, Q) and the set of squares of cosines of Q(J-, Q), where IF, Q and Q are nontrivial proper subspaces of Ti. i Theorem 4.3 Let P, Q and Q be the subspaces of a Hilbert space H. Then the following inequality holds: dist(cos2(Q(P,Q)), cos2(Q(P, Q))) Proof: By definition, cos2(0(Jr,Q)) = T,(PjrPg\^) and cos2(0(^r,Q)) = E(PyrPg\yr). Both operators Pj?Pg\? and PjrPg\jr are selfadjoint. Therefore using (4.1), we have dist&iPjrPMMPrPeW)) < \\PrPg\r PrPg\r\\. But IIPrPe\r ~ JVffcWI = IKPrPgPr ~ PfPgPr) Ml < ||PTPPT P?PgPr|| < ||iV||||P5 Pg\\\\Pr\\ < gap(S,g), and the assertion is proved. 88 5. Estimates for Discrete Angles Between Subspaces In this chapter we derive the estimates for discrete angles between subspaces. In the first section we generalize some results from [40] and [42], First, we esti- mate the maximum of the difference of cosines of the angles between subspaces, and next we obtain different results on the weak majorization of the angles. 5.1 Some Properties of s-Numbers that Are Useful in Obtaining Estimates In this section, we give some known properties of s-numbers of a bounded operator that are generalizations of similar properties of singular values of ma- trices. For these properties see, e.g., [5, 32, 33]. For majorization see, e.g., [5, 33, 53]. As we already mentioned above, from the definition of s-numbers it follows (see [22], p. 61) that for any operator A E B(7i) Sj(A) = Sj(A*), j = 1,2,... (5.1) and for any scalar c Sj{cA) = \c\8j{A), j = 1,2, (5.2) The well-known result on an approximation property of the s-numbers of completely continuous operators can be extended to the s-numbers of bounded operator. Namely, the following theorem and its corollaries are true: 89 Theorem 5.1 ([22], Theorem 7.1, p. 61). Let A Â£ B(Tt). Then for any j 1,2,... ^(^)= mm ||A-K\\, (5.3) JA where Â£j_i, j = 1,2,... denotes the set of all finite dimensional operators of rank < j 1. Corollary 5.2 ([22], Corollary 2.2, p. 29 and p. 62). Assume A,B& B{TL). Then for any i,j = 1,2,... the following inequalities Sj+ji(A + B) < Sj(^4) + Sj(B) (5.4) and Si+j-i{AB) < Si(A)sj(B) (5.5) hold. The following inequalities are also true for all A,Be B(H) and for all n = 1,2,. (see, e.g., [22], p. 63): n n n Â£>,(.4 + B) < Â£>(.4) + 5>(B) j=l j=1 j=1 (5.6) and 71 71 For j j=1 1 = 1 and z = 1,2,... we get from (5.4) and (5.5): Sj{A + B) < Sj(-^l) + ||-B|| (5.7) and Sj{AB) < Sj(i4)||B||. (5.8) 90 5.2 Estimates of Absolute Error of Sines and Cosines Here we estimate an absolute error of cosines/sines of the discrete angles be- tween two pairs of subspaces. One subspace is fixed in both pairs, and the other one changes. We estimate these quantities by the gap (aperture) between the initial and changed subspace. Next, we consider the case, when both subspaces of the initial pair are changed and we get the estimate, using both gaps. Let us denote by gap(!F, Q) a gap (aperture) between subspaces T and G (see, e.g., [2], [22], [37]) which is defined by gap(^g) = ||PT P0|| = max{||PF{I Pg)||, ||P0(/ P*)||}. (5.9) The following lemma gives an estimate for the proximity of cosines of the smallest angles when only one subspace is changed: Lemma 5.3 Let 9\ and d\ be the k-th smallest discrete angles between the sub- spaces F and Q, T and Q, respectively, k = 1 = min{dim T, dim(7}- Then for k = l,...,q, | cos(^)^-cos(^j]^_2 gap(Q, G), (5.10) where k2 = max{cos(0m;n{(Â£? + Q) G,F})\cos(0min{(Â£ + G) QG,f})}- (5-11) Proof: The proof is based on using the inequalities (5.7), (5.8) for the identity PrPS = PtPqPq + PAI ~ Pg)Pg, (5-12) where I is identity operator on H (see [42]). 91 (5.7) and (5.12) give: cos01) = St(PrPg) < St(PfPePe) + \\Pj:(l Pg)Pcl (5.13) But PrU Pe)Pg = PrPfgZfiggV Pg)Pj and we have IIPAI ~ Pg)PcII < l\PrP^)eemi ~ Po)PgII- (5.14) Using the equality (5.9) we get from (5.14): IIPAI Pg)PgII < WPrP&^egho.riQ, 6). (5.15) Next, using the inequality (5.8) for the first term of the sum in (5.13) we get: Sk{PjrPgPg) < Sk{PrPg)\\Pg\\ < Sk(P?Pg) = COS($1). (5.16)
Prom (5.13), (5.14), (5.15) and (5.16) the inequality
cos(^i) < cos(^) + \\PrP^-)eg\\gap(g, Q) (5.17)
follows.
Changing the places of Pg and Pg we get instead of (5.17)
cos(0j) < cos + \\PrP^-)eC\\gap{g, Q). (5.18)
From the definitions of s-numbers and angles between subspaces we con-
clude that WPrPfrffiogW = ^(PfP^TSieg) = cos(0min{(C? + G) G,P}) and
\\prpJ^)eg\\ = = cs(0min{(Â£ + G)QG,P}). These two equal-
ities together with (5.17), (5.18) and (5.11) give (5.10).
An estimate for the proximity of cosines of the smallest angles between the
subspaces when both of them are changed is given by the following theorem:
92