A theory of cross-spaces (Annals of mathematics studies series;no.26)

A THEORY OF CROSS-SPACES BY ROBERT SCHATTEN

ANNALS OF MATHEMATICS STUDIES Edited by Emil Artin and Marston Morse 7.

Finite.

Dimensional Vector Spaces, by PAUL R.

11.

Introduction to Nonlinear Mechanics, by N.

14.

Lectures on Differential Equations, by

15.

Topological Methods

HALMOS

KRYLOFF and N. BOGOLIUBOFF

SOLOMON LEFSCHETZ

the Theory of Functions of a

in

Complex Variable,

by MARSTON MORSE

CAHL LUDWIG SIEGEL

16.

Transcendental Numbers, by

17.

Probleme General de

18.

A

19.

Fourier Transforms, by

20.

Contributions to the Theory of Nonlinear Oscillations, edited by

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Functional Operators, Vol.

22.

Functional Operators, Vol.

23.

Existence

24.

Contributions to the Theory of Games, edited by A.

25.

Contributions to Fourier Analysis, by A. A. P. CALDERON, and S, BOCHNER

26.

A

27.

Isoperimetric

la Stabilite

du Mouvement, by M. A. LIAPOUNOFF

Unified Theory of Special Functions, by C. A. TRUESDELL

S.

S.

BOCHNER and

K.

CHANDRASEKHARAN

LEFSCHETZ

Theorems BERNSTEIN

in

I,

II,

by JOHN VON

NEUMANN

by JOHN VON NEUMANN

Partial

Differential

Equations,

by

DOROTHY

W. TUCKER

ZYGMUND, W. TRANSUE, M. MORSE,

Theory of Cross-Spaces, by ROBERT SCHATTEN G. SZEGO

Inequalities

in

Mathematical Physics, by G.

POLYA and

A THEORY OF CROSS-SPACES BY ROBERT SCHATTEN

PRINCETON PRINCETON UNIVERSITY PRESS 195

COPYRIGHT, 1950, BY PRINCETON UNIVERSITY PRESS LONDON: GEOFFREY CUMBERLEGE, OXFORD UNIVERSITY PRESS

PRINTED

IN

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TABLE OF CONTENTS Page

INTRODUCTION 1

.

2. 3. 4. 5.

Statement of the problem

1

Purpose of this exposition Acknowledgement

3

6 6 8

Plan of study Outline of results

NOTATIONS AND CONVENTIONS I.

THE ALGEBRA OF EXPRESSIONS 1.

2. 3. II.

1.

The normed linear spaces Crossnorms The bound as a crossnorm The greatest crossnorm The Banach spaces The inclusion (

all

operators from "^, into

a^-norm which may be approximated

of finite

Finally

is

B^T?^_)

be considered as the Banach space of into T^

.

normed

c~norm

of finite

For every operator

settle the "extension*

A

we have

problem

for Jj

in that

||AJ|

=

BS^'l^z

"pj^

if

it

rnay

(from T^i^

norm by operU|A|||.

by proving that in

1

INTRODUCTION

2

general * and only

is not of "local

TrC

if,

C

W*^C

manifold ~Y$L C, T^

* or

T^B^x*^-

space

is

T@>

unitary

if

every two-dimensional linear

.

In Chapter IV, a

termed an "ideal"

A Banach

character".

j^ft

K&

Banach space

if, (1)

of operators

A

together with

l

.

from T^

YAX e

also

4| is

^

into

for any

1ft

,^k

X

pair of operators (ii)

)(

YAX

J

in

.

||

^CH(X|||

Y

and

on Tp and ^-^respectively, and f

|||

Y||

||

For a crossnorm

A|| o
\

.

non-zero terms,

of

n-dimensional or a Hilbert space. Furthermore, we

uniform crossnorms, that A(|

number

finite

of finite rank

and any pair

crossnorm. Every unitarily invariant crossnorm

define the Schmidt-class

which 5?

on fv

f

the last

sum

A

and

r

B

is

.

A (D ^

||

(sc)

,

!,

\Jty-

trace-class and also the space of

ail

completely continuous operators on a Hilbert space are non-reflexive. Finally,

we introduce

tary relations which they satisfy.

corollary

we deduce

mined by

the values

natural In

assumes

number smaller than Appendix

II,

In particular, they are all reflexive.

we present a

term "self-associate"

we

*s

fi

As

a

no * deter-

(where p

is

any

the dimension of Fv)> definite construction (not unique however), Tft

,

1g

furnishes a definite crossnorm on 1^

unitary spaces

py

for operators of rank 4? p

which for any two Banach spaces

justified to

crossnorm on

for instance, that a it

some elemen-

"limited* crossnorms and discuss

since,

,

(without any special restrictions!),

O^^.

The resulting crossnorm we are

when our construction

obtain the usual self-associate

crossnorm

is applied to

(f

on

*ft

1

NOTATIONS AND CONVENTIONS

6

We

assume

shall

that the

reader

is

familiar with the elementary con-

cepts and theorems in Banach spaces and in Hilbert spaces, as can be found in

l]

and

The

l8]

definitions and

First,

gories.

.

theorems throughout

we have theorems which apply

this

paper

formulate in the most general form.

to

two cate-

to perfectly general (and

times only reflexive) Banach spaces, hence equally well

These we prefer

fall into

to unitary spaces.

They form the con-

The other type

tent of the first four chapters and of both appendices.

some-

is for-

mulated only for unitary spaces. The symbols of LI, p. 26j

,

feJt

while

>

will be assigned to two linear spaces in the sense

Vv^

and Vv will stand for the linear space of

Vv

all additive

(which in our terminology will also imply homogeneous) numerically valued functionals

[l

,

on y

p. 27J

Banach spaces, that

is,

and

VvL

1& and T^Lwill stand for two

respectively.

two normed complete linear spaces

and 1^ will stand for their conjugate spaces

Banach spaces and

*T^

of all additive and

bounded

[l

,

l

,

p.

188]

jTl

,

p. 53y

,

will be

termed

**

equivalent"

if

while

that is, the

pp. 54-55J functionals on Tji

respectively, where the bound of a functional represents

Banach spaces

,

its

norm. Two

they can be transformed into each

other in a one-to-one additive and norm-preserving fashion, in the sense of [l, p.

180].

Py will stand for a linear space in which there (



also be represented

normalized

shall denote

,

A( af + bg

and any constants a

whose domain

will be reserved for operators.

the sense of

p,

p. lOOj


A

same =

)

b

,

v

,

"additive and bounded operator**

contained in the if,

.

of definition is the

Banach space.

or another

aAf + bAg

The letters

for any pair of

A

,

B

,

A

will denote the adjoint of

when considered on general Banach spaces.

C

It

X

,

in

should

be understood however in the sense of Qs, Definition 2.8J when considered on unitary spaces. "finite rank**.

||A|)

An operator whose range

The symbol

stands for the

norm

of

A

is finite

dimensional

is

bound

A

will represent the

)|)A|||

,

when A

is

of

termed ,

of

while

considered as an element of a

Banach space whose elements are operators with a norm not necessarily equal to their bound.

For a Hermitean operator A on Pv

,

Pv

that is, additive and bounded transformation,

whole space and the range

F,

,

orthogonal sets (nos) or complete normalized orthogonal sets (cnos) in

and sometimes in (closed linear) subsets of

7

.

k^

,

g^

*$*, while

t

and

By

T^ and Olt^will

v-

S^T^

g^

.

Among

subject to the following rules: f

^? 8^

denotes any permutation of the integers .+.

fjig^*

these

+

yg^.

1

,

2 ,,..., n

20

THE ALGEBRA OF EXPRESSIONS

I.

DEFINITION

be termed equivalent,

shows fhat

(i)

^^

(i)

Some elementary

(ii)

,

(ii*)

,

is reflexive, that is,

The definition also implies

lent to itself.

and

g^

f^.

-

ZlTh.a

will k^

one can be transformed into the other by a finite num-

if

ber of successive applications of Rules

Rule

2^

Two expressions

1.1.

Z^f^ g^

,

We

(iii).

write this,

every expression

is

equiva-

transitivity.

For instance,

results can be readily obtained.

if

then,

>

h

LEMMA

1.1.

Every expression S.cl

or to an expression k,, .....

,

2-^

k^-

^s

8i

^c!

in

h. /

t

equivalent to either

which both the

h,

,

......

,

h^

and

k,^, are linearly independent.

Proof.

Suppose that

in either set

f

ments are linearly dependent. Then, 2T?^ sion involving only 1,9 g,

.+

We may h either 2fT', J 85 1