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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

21.

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

THE UNITED STATES OF AMERICA

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
2. 3.

(

4.

The space

5.

6. 7.

19 21

fi^0 fe*

22

T^^otT^a. an<*

A

^9i

<

T^f^U.'

^*

C ofC^au/^0*"of finite ot-norm

as * ae space ^t'*^.) of all operators

25 27 30 36

^,U'9v and U,7?

^??^UT?J

operators

-natural* equivalence as a space of operators The **local character* of "y as a characteristic property of unitary spaces

.

"9(^01,' "9T

42 43 44 47 48 49 53

IDEALS OF OPERATORS 1.

V.

Transformations on expressions

gc

CROSS-SPACES OF OPERATORS 1.

IV.

2l7:,f c

The expressions 2EL~.,fc gc. The linear spaces Js^Oft^ and

CROSSNORMS 2. 3. 4.

III.

16

60

Ideals of operators

CROSSED UNITARY SPACES 1.

Preliminary remarks

2. 3. 4. 5. 6. 7.

The canonical resolution

A

for operators

characterization of the completely continuous operators

The Schmidt-class of operators The trace-class of operators Symmetric gauge functions The class of unitarily invariant crossnorms

65 67 69 72 77 84 93

TABLE OF CONTENTS

vi

8.

9.

& ^^ )*" The space The Schmidt-class of operators as the crossspace fC
(

10.

space 11.

12.

2. 3.

= (

*>*)*_ (

APPENDIX 1.

et<^"R

The structure of ideals f 8^ PC )* A crossnorm whose associate is not a crossnorm

110

114 123 128

I

Reflexive crossnorms Reflexive cross-spaces w

108

L,imited w

APPENDIX A 1.

crossnorms

134 138 143

II

"self-associate

REFERENCES

**

crossnorm

148 151

A THEORY OF CROSS-SPACES

INTRODUCTION

Statement of the problem.

1.

linear vector spaces (2 and {^, the well-known concept of their

For two

sum*

"direct

t

in

symbol

fij

linear space of all pairs (

V

g,)

a(

^

8

Identifying

(

f

=

g

,

(*v

defined in perfect generality as the

is

*,+

and

f

.

f

x

^ or

)

&x

$

and (^are subspaces of

=(

a8

with

)

,

where fe

)

g t)

a^

(

)

,

f

(

+

$ (^

.

g

.

(5,

gJ

8,+

.

.

any constant ,

(

g

Thus,

)

a

with g + g

f

with the stipulations:

(2

is

,

we may assume

that S.

defined and the elemen-

tary rules of addition hold.

When

TJ^

and Ti represent two Banach spaces,

it is

customary

to intro-

duce a norm oC on "IJIG 1^ in a manner which assures that the sequence of pairs

(

is

f^, g^)

sequence f^ *

and

f

convergent in that norm to

g^

in the

> g

(

f

,

Banach spaces

With the last condition, the resulting "direct sum"

g

1^,

)

if,

and only

if,

the

and Irrespectively.

*9|^J*^^ S complete,

hence a Banach space. Together with 1^, 7?^' we consider their conjugate spaces

F

5l?i^,

G

G.

ship

9

(

f

TJ3?*.

,

g

= )

of pairs

(

Clearly, an additive bounded functional V* on

to a unique pair

corresponds

T& 1?*

and form the linear space

**, 7^,

<(

f

,

(

)

F +

,

G

)

<^(

F

,

G

)

for

^^t**?^

and conversely, subject to the relation,

g

= )

F(f) + G(g). This

correspondence

INTRODUCTION

?

is

The bound

clearly additive.

determines a norm

on

Ot

*?

of

such a functional

$ ^^

Ci!

=

JT

(

<x'(

)

F

,

G

and

always holds.

When space

(

*,

fj

&

(B

g)

>

and fv^are Hilbert spaces,

the inner product of the pairs (f

if

number

Fu,

+

g

(

f

^T_ becomes f

(

,

g

is

)

defined by the

gj

f

For Hilbert spaces

$

the operation

has turned out to be a powerful

tool quite often used, for instance, in the theory of closed

The theory

also a Hilbert

of the direct

sum

of

and adjoint operators.

two Banach spaces and

in particular of

two Hilbert spaces does not present any special difficulties and

is

generally

well-known.

These considerations suggest the following natural question: Given two

K

linear vector spaces

and t

v^

,

is it

possible to construct a "direct product**

that is a linear vector space say denoted with the

products

f

g

,

for

(

which multiplication laws, f

|+

*

(

^g

g

g,+ g z)

=

=

f,e* g f

*

+

s,+

ya f

*

symbol

lv.

Bf H/

f

formal

that is, the distributive laws g

,

gz..

hold?,

Furthermore, when a suitable (one or (ft.

*^3. a

Tfi

and "Ip^are Banach spaces,

more) norm

06

on

7^

Banach space. Moreover,

<5 i^>^^ma.y

be so defined, as to

the elements of

determine the additive bounded functionals on

1

we would demand

T5,Jft *9^.

that

make

should

gRi^. Their bounds

should

INTRODUCTION norm
furnish a conjugate

^

\s

^

T^

,

and the relationship

should hold.

when

In particular,

an inner product on Finally

we

Fv

<8

0v/

and Pt^are Hilbert spaces, we are able to define

Pv^ so as to make

it

this

R

should be interested in finding out whether the operation

$

so defined presents similar possibilities as It is

also a Hilbert space.

?

the purpose of this exposition to present a reasonable solution of

problem.

Purpose

2.

From

of this exposition.

the algebraic standpoint alone the notion of the "direct product**

for two finite dimensional linear vector spaces has been in a sense

by H. Weyl (25] and let

t^

(V-'Xwi

Let \r denote the space

.

of

denote the space of n-tuples (y

,

t

(y,

..--.yj

is

m-tuples

numbers (^....x^

The "product**

...,7^).

defined as the mn-tuple

of

mentioned

(x^

,..,x^

f

;...;x

(

yw

,

..^yj

.

For these products the distributive laws hold. The products alone do not form a linear manifold.

by

T^

&

\^

,

However, the set

fills

the entire

mn

interpreted as a matrix of rank as an operator.

Thus

of their finite

sums, which we shall denote

^

dimensional space. Such a product 1

with

m

it

be

rows and n columns, hence also

for finite dimensional spaces,

^

^^

essentially the space of all rectangular matrices with a fixed

and columns, hence

may

represents

number

of

rows

possesses a well-known algebraic structure.

But even when the linear spaces

yi

,

fed^,

are infinite dimensional, a

INTRODUCTION

6

3.

Acknowledgement.

At

this point

with Professor

The in

J.

Institute for

it

seems proper

von Neumann

acknowledge that the author's discussions

1944-46 (during the author's membership

Advanced Study) followed up by an exchange

1946-48 have played

Needless

in

to

a decisive part in

preparing the foundation for this draft.

Professor von Neumann and the author. Some

full

form)

in their joint

the merits and credits

Plan

it

p

and

q

denote by

,

While the authors assumes

and [l6j

may

be contained in this exposition,

be shared with Professor von Neumann.

to

expect, suppose first that

dimensional Banach spaces, whose elements

F and G

*p^_

The expression

^.7^ F(f^)g^ can be obtained

3*

denoted by

_

^

X-

f.

^

gu

into T?^ (or G(g)f

may

f Si

g

for instance, as

froml^

into

^>

)

of

rank

intol^. Furthermore, every operator from

The algebra

obvious algebraic laws for operators.

The

last

1.

-$

be interpreted similarly, as the operator

such a manner.

such expressions.

we

^

particular the distributive laws of the symbol

of

will be

.

from

from in

i^o.re finite,

denote their conjugate spaces whose elements

As was mentioned before, we may then interpret F(f)g

^^ and

1*

while T& and

the operator

were published (although

of these,

of study.

)

and g

f

[l 5j

may have must

To have some idea what say

papers

responsibility for the shortcomings that

4.

correspondence

say that the ideas contained in this draft were originated by both

to

in different

of

at

may

for these

^

^

expressions and

into

T?

in

are determined by the

"Ve denote by

*)&

Jfi

^.^the linear space

clearly be identified with the

pq dimen-

INTRODUCTION from

sional linear space of operators

Defining on

it

"direct product"

ever O6

a

norm

^,

construct the linear space the double

^^"Fj

G^-

bounded functional

(this

that is,

T^

is

<X

& ^?V

^

determines a normoC

on

"13

g

(2

T^^

=

)

-

6^^

(

f c

(

-?,

ek^ j*

=

H g

f U

ZE.

Similarly,

.

Go

f

.

For

we

a fixed

represents an additive

bound

Its

.

II

7TT F o a

)G (g c-)

"associated"* with

every additive bounded functional can be obtained have

II

expressions

expressions on 7^

of

f (S

(

obtain a

also termed a "cross-space" when-

^J^ ^ L=I F

sum

many ways) we

can be done in

The last

*,^?V

"cross-norm"

is a

O6

TJ\ into

c

<x'

5f

(

.Tj"

F

0$

G

')

Moreover, since

.

such a manner, we

in

*<*

Guided by these considerations we define a direct product for general

Banach spaces.

The situation

is a trifle

simpler for reflexive spaces. In that

case again as before, the linear space of "expressions'*

may

be interpreted as

\s

the precise space of operators of finite rank

from

*)^

into

"^

.

For

the non-

reflexive case however, the set of expressions determines a proper subspace of the linear

approach

is

space

of

operators of

finite rank.

In that

case the most general

desirable by considering "formal products** and "formal expres-

sions" subject to the rules suggested by the previous case. of it

expressions a

norm

oiy

be complete.

.

is

Once

a linear set

constructed, then following the previous pattern we define on

In

general however, the resulting normed linear space

Whenever

this

occurs we "complete"

it,

that is,

imbed

it

may

in the

usual Cantor-Meray fashion in the smallest possible Banach space which shall denote by

norm

Ot

T^

L,^3*

^s

De f re the

norm

on the linear space of expressions S-.-. v

not

we

determines an associate

06 F.

O

<8&

G 13

.

The last space

INTRODUCTION

8

when completed furnishes

the "associate space"

we are already confronted

with the following interesting phenomenon.

jugate space for a given cross- space

may

associate space as a proper subspace.

well as the characterization of

some

^3^ (tt^^r^

Here however,

.

The con-

(and generally does) contain the

Finding out their exact relationship as

of the

cross-spaces their associate and

conjugate spaces, is one of the main problems of the present exposition.

5.

Outline of results.

In Chapter

I,

we consider two Banach spaces For

special restrictions. f

^

g

.

e. "3

sums)

51^,, fjS

algebraic elements, yet

we

by means of "operators of

g

Tc^

^^

without any

construct "formal products"

^

shall often use for finite

A(F)

,

*

^cK.

"formal expressions'*

g. Although these expressions are abstract

=

them a

specific representation

rank". Indeed, an expression

be interpreted an an operator

by means of the relation

we

"

,

With these, we form a linear set

(that is, finite

may

f

^

A from

ti& gc

into 1?^ of finite rank, defined

>5>

JEL^, F(f c )g c

-2ELc^ (

.

In particular, expressions

which furnish the same operator are considered "equivalent" and are combined into a single element.

For these expressions we set up algebraic rules as

example, the one expressing the distributive laws of

of the

&

fo

symbol, addition

expressions and multiplication of scalars by expressions. These are sug-

gested by the algebraic laws governing operators.

Next we consider some

algebraic relationships between these expressions, since the distributive

property of the

symbol, introduces certain linear dependencies.

Together with ^^

T?k^

we similarly form

the linear set

T^O"^^

of

INTRODUCTION expressions ZTTj F^ * * sions (

M&

5^i

Sjw'

F.

&

G:

IE a|

^'X S-T^i fj* (

on

1^

we

II,

T^_ and

obtain a

linear space

the last ones

we

^, Sf

pair of operators

S

For on

tional

(

*s

2^

Due

,

"norm"

"i

is

proven.

oc(^^

iJ8

ot,

(

f (8

=

g

)

f

II

II

H g

II

is,

for any pair

C

(S

T 8c') T

and

^

|I|S

M'

Ml

^

T l"

<^^. fc^

SL)

f

,g

for an ^

on "^ and 'T?^ respectively. The significance

norm

defining

K

G,)(

on

o

Gt'

S^,

(

O T^

TJ3>

to their significance

as the bound of the additive func-

F. Ql G-) O

^.^7 o

of expressions

gj

f,.

we construct an "associate" norm

Z,lt

that for any two

i

on

two uniform crossnorms are singled out and "y

and the bound

The

last one is therefore of "general character*.

also the crossnorm

A

where

the operator 5" J^]

character", that

,

F(fjg^ from

is, the

relations between the

A

Banach spaces, a unique greatest crossnorm can be

always constructed.

ing

g^)

This can naturally

*"?Su

discussed in detail, namely, the greatest crossnorm

We show

number

discussed later.

a given

^. OT^

of the

single out those which satisfy the "uniformity condi-

tion", that is, ct(

of this class is

expres-

interest however are the "crossnorms", that

many ways. Of

those which satisfy the condition

Among

means

defined by

normed

of

an Dinner product"

define in the usual fashion a

Chapter

be done in

For a pair

).

f)

whose invariance under "equivalence"

2l.7^-5L^ F, (f^)G.(g.) In

*

F.G^-

is

8v)

^

G

and

.")

)

and

gc

F

(for

**

9

value of .*s

^

^ ^

(-^V^

into

(

1

a

S.J1T,

^ .

is

represents the bound of

8c)

Moreover,

y^f gj

So

"X

is

also of "local

depends only on the spatial

and those between the g Js, and not on the includ-

them spaces. We establish

that

^

is the least

crossnorm whose

.

INTRODUCTION

1

--- ^^..

Xf

is also a

^

7^

06", at

,

we prove

Finally,

.

associate* o is,

*\

*

crossnorm. Furthermore,

when defined on

be considered as the associate with the greatest crossnorm y

may on

w^

whenever

that

also a crossnorm, then

is

for a

crossnorm

&.

if

^

O>^,

defined

at its "first

associates of higher order, that

its

are~also crossnorms.

.....

In Chapter

we

III,

*'

complete* the normed linear space T^ Ot?r^_in the

usual Cantor -Meray fashion f6, p. 106]

,

that is,

imbed

it

into the smallest

possible Banach space by considering the space of fundamental (Cauchy) se-

quences of expressions (that

elements which they represent)

is,

in

^ ^L^rV.

and introducing some standard identifications. The completed space

Banach space which we

shall denote by

fi^T?^

TgS,

.

is a

The last space which

naturally depends on *b is defined as a "direct product" of "^ and Tfl

whenever

ot is a

crossnorm also as

od^^

For a crossnorm The

finite

the additive functional

1> B^Tr^.

fore also on .

Thus,

ing on the particular

sideration, is in Ok

,

the

(

T?i

SJ^

Fj

(

tfte

0"?^ F.

-^^

O^)

ij& gj

G^)(S^

t' is

also a crossnorm on

represents the bound of

on

Tjt,

0^^

,

there-

Completing T^ CD^*^. we obtain the "associate space

U>

z.)

15

Banach spaces

many cases

T*'^. The last inclusion depend^x and crossnorm c6 under con,

*9j,

a proper one.

Therefore, for a given crossnorm

cross-space "^ JS^"^^ deter mines uniquely a conjugate space

a (

(

ot'

or

a "cross-space".

on T

number

,

l?}^^^.)

it

and an associate space T^

^ ^st^^rL..

Although we have a fairly

good idea what the associate space of a given cross-space represents, we do not find

it

easy to state the precise restrictions imposed on a crossnorm &' for

INTRODUCTION which the resulting cross-space with

^i^v"?^

the greatest

^

w

from

"9j

into

1^

(from"^into

norm. On

sents its

which

Incidentally,

may

we

^

)

the other hand,

preted as the Banach space of into Tf*\)

crossnorm

.

Jf

In the last case,

be characterized as the Banach space of

may

)

such, that its conjugate space coincides

complete discussion has been presented however,

when we deal with

for the case

(

A

associate space.

its

is

1 1

all

where the bound

K^

fiL,/

TJ^

=

be approximated "in bound

1*

operators

of an operator

repre-

gf^Tj^ may be inter-

7^^

^ into ~^^.

operators from

all

(from *^lu finite rank.

by operators of

establish a "natural equivalence" between the space of all

(those which can be approximated "in bound** by finite rank) operators from y*>

by

and the space of

into 1?k

finite rank)

Let

operators fromlj^into '^

be a given crossnorm on

C

jf

C

sion

such that

5Ej2^ fJ8

always have

|-5

^

A

||^

^\\\

A

^

Tr*x

.

An

III

A from

operator

\

^

CQcf^^,

For any crossnorm

.

finite

TP> t

con-

fjA g^) for every expres-

of such constants is denoted by

&. The least I)

Af^Jg^

(

4

.

|

ot-norm* whenever there exists a

into y&^ is said to be of "finite

stant

which can be approximated "in bound*

all (those

O6 ~&

A

It

,

A

11^

.

We

the space of

&.

operators

!?, into

represents the

|lA Ij^

may

A from

be interpreted as

"^^

norm (

")S^

of

A

)

ators of finite rank.

we

a

This space

is

complete. Moreover,

On

the other hand,

linear space

"3> (

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
,

where

the

,

norm

A|| stands for the

||

Banach space

of

operators of

A

of

finite

.

O6 If

-norm,

that is,

(

T^ Sj^TJi^

QI is uniform, OC is such, and In

Chapter V, we make use

in addition of the tools at finite

an ideal

is

)

forms an

p^ of

and only

if

ideal.

our previous results and avail ourselves

our disposal in unitary spaces

f\/ (in

interpretations of the direct products of unitary spaces,

decide to interpret ff/G finite rank,

f\,

and denote a crossnorm accordingly by

<^,(

X

A

written uniquely in the "canonical" form (**J.)

form orthonormal

whenever the sum has an

on p^ of

infinite

sets, the

number

of

point proper values (with multiplicities) of

).

We

single out

=

SL^, a t*fJTpc,

a^'s

are

>0

terms. The abs(

A

A

which

we consider

the unitarily invariant

be

where both

and lim a^'s

may

a.

form

=

the positive

).

Following a brief discussion of some properties of symmetric gauge functions,

we

and characterize them (hence also

ft/

the operators of finite rank) as precisely those operators

and

X

as the linear space of operators

the completely continuous operators on

VJ^)

the sense,

dimensional Euclidean spaces or Hilbert spaces). After some brief re-

marks on possible

(

QL is uniform.

if,

crossnorms OC on fv

fC

INTRODUCTION that is, those which satisfy the condition

X

operator

We

of finite

=

o( UXV*)

Ot

(

X

rank and any pair of unitary operators

from the class

for any

)

U

actually characterize the precise class of unitarily invariant

as the one which can be generated

5

1

V

and

crossnorms

symmetric gauge functions

of

on the linear set of n-tuples of real numbers or on the linear set of infinite

sequences of real numbers having only a depending whether

f\/ is

prove that the class of

(II

||/B

A

ators

Q6(

|||

and

X

)

B

crossnorms coincides with

of unitarily invariant is,

those which satisfy the condition

X

for any operator >\

.

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)

,

<.-+-

=

OC*

oo

the

sum

A (P.

2L^(

,

The

B

convergent and independent on the chosen cnos

is

an Dinner product" on (sc) and

norm

that goes with

it.

(

A

that is, the

,

A

Banach space

OC*

is

A

We

orthonormal set ((^

=

Cp^ (

)

(#)

(

A

,

S*(

X

)

prove that

)

is absolute-

The number

is a

= 0*

all

may

and

A B ,

(

d(

crossnorm on

of all operators of finite

be characterized as the Schmidt-class; they

B

);

For

is a linear set.

(sc)

which we define as

)

In particular,

linear set of operators of finite rank.

is also a

identically.

for a complete

,

ly

the

oper-

as the class of all those operators

(sc)

^ j)

O(,

independent on the chosen cnos. in

of

represents the least unitarily invariant crossnorm.

reflexive, that is, satisfies the condition

We

^

o<,(AXB)

Consequently, the associate with every unitarily invariant crossnorm unitarily invariant

the class

(

A

)

)

is

the

p^ ^j^fv

= /

3 -norm, may

be approximated in that

1

INTRODUCTION

b

norm by operators

of finite rank.

Next we show that the linear space

of all

completely continuous oper-

ators on fv where the bound of an operator is considered as the

Banach space

f(,

associate space and is the

for a

Banach space

Its first

&^f\,

of all

A

operators

complete orthonormal set

(

^)

on

F\s

,

for

and where the last

Banach space represents

of all

its

sum which

crossnorm

is

(

py

*>^*

norm

of

is

-

indepen-

A

w

-^

may

)

be interpreted as the

norm.

we discuss

v $L Pv

A crossnorm (^-norm

(

(abs(A)
operators on Pv where again the bound of an operator

In this connection

f^V&^Pv and

T5

its

The trace-class

which SL^

dent on the chosen complete orthonormal set represents the

The second conjugate space

norm, furnishes

conjugate space coincides with

be interpreted as the trace-class.

rjnay

its

* or

OC/ is

anv

**

termed

the topological equivalence of the spaces

limited* crossnorm OU **

significant*

if

.

every operator

of finite

completely continuous. For any significant unitarily invariant

pv && v)

rnay be characterized as the Banach space of exactly

all

those completely continuous operators in the canonical form 2IL^ a^C^Ol~f

for

which lim

sents the

(

norm and

Finally

norm

6*

S^a^ifjtf.^ )< equals ||A

we show

-f

oo

.

jj^.

^

does not represent the least crossnorm on py Ofv

The constructed crossnorms which are not of

of the last limit repre-

that for an n-dimensional space Pv/^with the Euclidean

(or Hilbert space),

amples

The value

,

^^

at the

same time furnish ex-

crossnorms whose associates are not crossnorms. They may also

serve as examples of crossnorms which are not unitarily invariant.

INTRODUCTION In Appendix

I,

we present some scattered

For the sake

gations.

reflexive, and all

of simplicity

crossnorms

Q&,

1

results for further investi-

we assume throughout

We

^ 7^

are

=s

(ii)

have always, 0L

The following conditions are equivalent:

Ot**OC,,

^

holds.

t

(iii)

ot*(3'

for

some

/3

We term

.

Ofr'=

(i)

implies ot^/9

/

reflexive whenever

of,

(i)

crossnorms and prove the equivalence

a cross-space is reflexive,

associate space

(ii) its

of re-

of the following statements:

conjugate space of the cross- space coincides with

its

is reflexive, (iii) the

associate space, and the

As an

conjugate space of the associate space coincides with the cross-space.

example, we also discuss the non-reflexive cross-space *

+ -L

a

J

(ii)

always reflexive.

is

Next we investigate reflexive cross-spaces generated by means flexive

and

.^C

) .

<j

*^ and "^^are

that

.

OC

5

.

in particular both, the

where p >!,

\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 (

^

)

and which

is

separable and complete

l8,^2 ]

is

.

defined an inner product

Thus, Pv

is either a

Hilbert space or a finite dimensional Euclidean space according to whether contains an infinite or finite

number

of linearly

independent elements. R/

it

is the

NOTATIONS AND CONVENTIONS space obtained from (

^

,

J,

in

which the fundamental operations Ct^ ^P+'f

are replaced by Su^p,
,

)

f$/

^

for closed linear manifolds in

Elements in

in

"

H.

g

V ^ >

^

an<^

<4^

>v

&*

py

(

Tpi (p.

)

,

.

Elements

,

(

^is,

)

,

in

we

,

f

,

The

.

G

'

,

stand

K

0v

h

f

,

.

F

letters

,

G,t

w^ill

h.

,

H

,

K

,

Those

.

.

,

u

The term

operator** will always

We

elements

Y

A

recall that

,

f

,

g

is

is additive

mean

will be

i>

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

<0>
(

T^ or

k

,

'

assigned to elements in by

and

,

fv will be denoted by

73. or

will stand for elements in

,

u

*

v are given by

^v 01

(v^

K,,

1

(that is

such that

A*

=

A

)

we

,

1

NOTATIONS AND CONVENTIONS

8

shall use the

when

U

,

(

V

A

<jp

We

.

term *P

,

"definite* in the sense of non-negative definite, that is, is

)

on

equal to

its

which

W

,

the

"Wl, is

symbols

i

j

,

3

and

T^a

m

,

,

termed the

initial set of

n

,

p

a

,

a.

,

q

.

,

b

,

b^

The abbreviations

W

inf

and

;

integers

For a complex number

will stand for its real and

a

and

"K/L

its final set.

denote complex or real numbers by

part respectively. M

for a partially isometric operator

on a closed linear manifold

isometric

is

also termed

is

will be represented by

a

W

orthogonal complement;

while the range of

We

Unitary operators will be denoted by

.

shall reserve the letter

that is, one

on Pv

^

always

sup

imaginary

will stand for the

greatest lower bound* and the "least upper bound** respectively. 1

We of

denote by

expressions

,

A

2E^f;fl g^

crossnorm, while Ct* is

ot

defined on

,

.

....

crossnorms on

In particular,

^

the linear space

T-bOT^i

will stand for the greatest

A represents the crossnorm furnished by the bound. "V^

Cantor-Meray closure

T^ of the

and denotes the

normed

norm

associated with 06

linear space "V^

O^T^^

is

.

denoted

The

,

19

CHAPTER

I

^^^

THE ALGEBRA OF EXPRESSIONS The expressions 2Lci IL&

1.

this chapter

Throughout spaces and

,

jij

p~

,

gc

8L*

we assume

that K? and two Vv^denote any

linear

(

denote the linear spaces of

all additive

functionals on (,

and ^^respectively.

We fj,

,

in

f^

(v,

and g

,

,

*,# g,' +

We may

#

introduce two symbols

-

6X

g

in

V*

8^'

+

+

^08,-+* y> 8 X

(i)

;

1

(u)

2

,

(

; ,

f,'+ fjj

,

e

n;

g;

'

+-

****

'

+

With these for

.

we construct formal "expressions" +'

'

abbreviate the last expression by writing

expressions we introduce a relation

where

and

f

^?8>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

n-1

2:r.vfu

......

f ,

f

f^

,

8c

terms. For instance, c

^(r^Vt)

8,

or

f^ ^

if

+

s

f

g t , ...... ,

g^

the ele-

equivalent to an expres=

2l c*^ 3^ c

then

ZLta^t* 8C

therefore continue to reduce the number of terms until we have k^ v

in

which both the

...... , h., '

h AH^,

and

k

, I

...... ,

kare 1*4*

.

THE ALGEBRA OF EXPRESSIONS

I.

linearly independent or (Of)

^080

3

or

tf

DEFINITION form ^L^Jf^

g

1.2.

We

*

f

into a single

element

termed "an expression

fgr

permit "an expression for expression for a

and

we

define

T

+

It is

f

,

f

r>s f

".

*

t

27^ o

.

f,& i

g'.

v the set of

{?,0

expressions of

all

we consider equivalent expressions as

If

an expression

each other we

is in

f

Quite often we will find to stand for

f

If

.

an expression for

"S.^, ,

will be

convenient to

it

g

it

g^ is an

f^

then for a scalar

(3

as the set of expressions equivalent to

a consequence of Definition 1.1 and Rules

5T^

f **

8c

(

9 "

*"l

g<,

>

2jTi

(i), (ii), (ii*), (iii)

does not depend on the particular expression used. Similarly,

f

,

*j** 8^

that

f*

O/

af

(0.0) Sf

as the set of expressions equivalent to 2i>,(afc )

af

*g*

&

.

identical, that is, the class of all expressions equivalent to

combine

2t

.

and

denote by

In the last set

*

t

But

.

00

CK

g

R^,

t

g

(ft

&

Similarly,

.

The linear spaces

2.

the

f

*<* g c

-f

^^ g

is

defined uniquely. It is

easy

to see that the usual properties of addition

and multiplication

by a scalar hold; for instance,

r

+

a(T

9-

+

=

g')

r

+

=

aT + ag^

The zero element

THEOREM

r

?

;

;

+

a(bH

is the class of all

1.1.

with scalar operators.

|{v.

+

^> =

=

(ab)T

+

7>

+

^

.

expressions equivalent

is a linear set

,

that is

to

til

.

a commutative group

22

THE ALGEBRA OF EXPRESSIONS

I.

Proof.

F

A

,

......

we form ,

f

little

the linear set

are in K,

F^

more about

-

/B^ of expressions 2t\vT'F.

fc,

G |f ...... G ^

while

,

/v

fif

G

'

*

are in

).

these expressions, in particular the invariance of

their "rank" under equivalence,

we

g^

The proof follows from the preceding discussion.

Similarly,

(where

3Efj.

may

be found in

l2j

.

In the present chapter

only state those properties of expressions which will be needed in our

future discussions.

Transformations on expressions.

3.

LEMMA

C

F

Let

1.2.

Then,

?*. I

f.# ZT^ vSf i*

Proof. For a given expression JS'IT ^L^ 8,

^gJ the

same

(i)--(iii).

=

JE"J^,

F(fjg c

Since

.

for both sides of any of the

This

means

F

.

g-*^

^ 2^ h.f cl^i

we form

^

o

implies

the transformation

the values of

^** relationships

k.

4

T

remain

expressed by Rules

that a single application of these Rules to an expression

T

does not change the value of

.

Thus, a finite number of successive appli-

cations of Rules (i)--(iii) to an expression does not change the value of for that expression.

Similarly,

LEMMA

DEFINITION *f.fl|

g

>%

This concludes the proof.

we prove

1.3.

T

Let

1.3.

f

the following "dual" of the last statement:

.

. ,

Then,

Let^^RflTG,

JSE

FC

G^

T^i

be an expression in J^fe

represents an expression in 6,

S^

KJ im P lies

/* while

Under their "inner

THE ALGEBRA OF EXPRESSIONS

I.

product" in symbol (2:^j

8

F.

G.)(JE.^

(

^

JZT<

*

A

* gc.

23

we understand

g t)

the

number

^*,r,ww LEMMA mean

1.4.

The inner product

is invariant

under equivalence.

We

hereby, that K.

imply

Since SEf^ F 4T*' d

Proof.

(ZJ^

^ 1S ^X^S, Lemma

Similarly, by (

The

G/ ^

-

2JL,

"^

f

C

2T ? H. d*"' ^

C=s

K.

=

8.)

,

Lemma

1.3 gives,

*

C2.;.,

H.

V ^S, (

M

f

c

1.2,

V S^

f

(

(

C*

=

8 t)

(

2, H.

K,.)(

27.,

h

t

k.)

.

last two equalities furnish the desired proof.

The proof

of the following

lemmas

two

is

obtained analogously:

LEMMA

1.5.

Let

A

denote an additive transformation from K, into Ky

LEMMA

1.6.

Let

S

and

Then,

^respectively.

* ,

c

DEFINITION

T

Then,

^2, v* v 1.4.

denote two additive transformations on

impiies

^, ^ T *C^ -s.^ sh^ T sf

For a fixed expression

define the following transformation

from J?

4

JJTj*

into

6^:

J*

8C

i 11

*!

2k

It is

the

THE ALGEBRA OF EXPRESSIONS

I.

a consequence of

same transformation

LEMMA Proof.

Lemma k

.'s

1.1,

Lemma

21^,^9

Sj^ g,;

=

implies

* c

is

g

F

~ -2^, h ^ ,

implies^\

G^for which

"

F(h.)k.-^

.

F(h

An

f

)

qk

Z

f

g^

f^fc

not equivalent to

where

are linearly independent. In particular,

find an

.

equivalent expressions furnish

1.2 that

.

Suppose

gt

A

of Definition 1.4.

T xr4

1.7.

Z,f

.

the

h^

ffc

<35

h.'s

.

of

.

Then by

.

as well as the

Consequently we can

The linear independence

application of

Lemma

1.2

of the

furnishes

k-'s

2^

and therefore,

This concludes the proof.

LEMMA alent

if

1.8.

and only

if,

Two expressions they furnish the

^^

f

J8f g

^ and yF?^ h .g k

same transformation

Proof. The proof is a consequence of

Lemmas

1.2

.

are equiv

of Definition 1.4.

and 1.7.

CHAPTER

II

CROSSNORMS The normed linear spaces

1.

^fiG^c'c^*.

Henceforth, we assume that T>and \f

while

"^

and

^^denote any two Banach spaces

^f

and

1^

stand for their conjugate spaces, that

is, the

spaces of

all

additive and bounded functionals on T^and*")^ respectively.

Clearly, an expression ^>

an operator

A

of finite

^

g^ in

f.

")

rank from *^ into

*^>

Zr

for

be interpreted as

may

T^L

whose defining equation

is

given, by =

A(F)

By Lemma

1.8,

also every operator

A

of finite rank.

of finite rank

many such expressions. To mined by where

assumed ajF)

=

may sum space

this

if

^

for

F

In the

case when *^j

from

T^Jinto 72

is

e

,

......

,

We

g,^.

">?".

elements

Thus,

f

,

.......

f

A

is

of the

then

determined by

A

A(F)

^y

f^

in

Tji

is

=

deter-

S^

Since "^ 1^.

determined by

considered as a subspace

they fur-

if,

is reflexive,,

have,

up as follows: For any two Banach spaces

^"f^may be

.

and only

represents additive bounded functionals on

to be reflexive, there exist

FffJ

fe

see this, suppose that the range of

the linearly independent

ajF)

F

two expressions are equivalent

same operator

nish the

F(f.)g t

and

is

such that

L lfc^ t

space of

A

all

g^

.

We

the linear

operators

26

CROSSNORMS

II.

from

1^, into

Ife

7^

finite rank.

Tj^of

^

case

In

is

represents precisely the class of

assumed

to be reflexive, then

operators from

all

into

*1^

"^

of

finite rank.

Throughout

and the following Chapters

this

and IV we shall assume

III

Banach spaces without any special

thatl^, and T^^represents perfectly general restrictions.

DEFINITION

Under a norm oC

2.1.

any non-negative function of expressions

'^^ we

in 1JS

2EI

^

f .(8

shall understand

g^ satisfying the following

conditions:

oC(aZ^

II.

* t

gj

c

and onlv

if

=

DEFINITION

F.

^

G/ O

1'

for all expressions JSTj

LEMK/IA then

it

2.1.

is

*

f-S g^ in

(^g

I

.

O

^

gc

^o +

Whenever

IV;

.

a

<*/(

.

cU2f)

21^ F. ^

For a

.

flT

G.) O

as

satisfying the inequality:

"^^.

ot' is finite

I

o

^T^i

Thus,

therefor* a function of expressions in "^f

also satisfies conditions

with 00

v

for any constant

we define * ** 1^'>?f;

in

the least (finite or infinite) constant

G.)

-2^1,

Let 06 denote a given norm on

2.2^.

expressionZ^ <J*

if '

Ul ^(-ST^f^gJ

^(rfg^Zferg)

m.

fixed

=

o6(2, *ja sJ

i.

">?v

for every expression in 04.'

is

^

termed the norm "associated"

CROSSNORMS

II.

Proof. Let

2j, F\ 8 =

G.)

F',......,

F' ^t

,

2?wr

obviously

IV

as well as the

F^

we can choose an

G'S

is a

G', ......

,

with

"^

1

.4,

>Q

,

we have Thus,

ZlT^agt.

2^ F & G.

F.

G^

GJ&*

2^

|

F^f)

^fe

.

with both

1.3, also

Since,

The linear indepena

g

e.1^such

^

jrjT ^(fjG^tg)

we must have cx'f^ 1^IB ^

'

G<

F^

is not

linearly independent.

F^(f)G^ 0. Now choose

By Lemma

.

T&-

<^(fjg|g) III

f

implies 2r?=|

F^(f)G^(g)

and

II

for every

.By Lemma 1.1,2^

dence of the that

.By Lemma

Suppose on the other hand that

equivalent to the

=

F.SLG^X^.f^gc) 2:7^, Fj

&

G. **

2?

F.-fiy f

o

.

Since,

>

G.)

3

.

are immediate.

consequence

of

Lemma

and the definition of 06

1.4

for a given oG.

This concludes the proof.

DEFINITION define a

define

norm <X

The linear set

2.3.

1^0 "^^

(Definition 1.2) on which

(Definition 2.1) will be denoted by

^^T^.

we

Similarly,

we

^1?^. LEMMA /^

implies

Proof.

2.

Let

2.2.

ot'

ou

^

The proof

ft'

is a

and on

A

represent two norms on

1f?^0

0^

consequence

T^7^

.

Then,

.

of Definition 2.2.

Crossnorms.

Among

the

norms oC on

"crossnorms", that

is,

norms

^ ^^

of particular interest are the

satisfying the following additional condition:

28

CROSSNORMS

II.

V.

<*(

Therefore, a

=

g)

norm

is a

f

I/

II

g

I/

IJ

crossnorm

and only

if

which generate operators from 1^

into

if, its

value for the expressions

^

l^of rank

equal to the bound

is

1

of the generated operators.

At

this point

make use

it

seems proper

of in the future:

given crossnorm ot

,

detail in the following

LEMMA of the

?

=

and the

fj,*s

f(t ;

llg.- g

.

l|

t

2.3.

..... .

V

<.

Proof.

,

We

j

for

In

T>

is

&^.

g, .......

and T^^are separable, then for a

IJi

also separable.

This

is

expressed

in

2.3 and 2.4.

A crossnorm ,

mention the following fact which we shall

case both

Lemmas

g^'s

to

<x(S^

B

,f

l r

that is, for a given ->

gj

such that

is a

gj

^

continuous function

we can

/|f t -f^i|

find a

,

implies

i=l,2,....,n

verify without difficulty the following relation:

2,f C B* 217,, > 8i ^ 2S,(c-^)8 t -+-2c;,'t(gt-8;) f

-+-sr:,(

i.-

f *)

Therefore,

M This concludes the proof.

LEMMA the

normed

2.4.

Let "^^ and T^^be separable. Then for any crossnorm /

linear space

'?j

|

^^,is also separable.

CROSSNORMS

II.

Let

Proof.

f.

f

,

and

,

g

g

.

is

f

denumerable and dense

in

2.5.

F

G

,'(

that for any pair

> F II

)

F f

,

REMARK <^'(

G

F

2.1.

oC on

ty

.

")j

It

F # G

)

we

It is

>

T*

F

G e !*.

.

a consequence of Definition 2.2

<x'(

F

G

)

Let

OG>

JB

o6(fg)

^

||F|/

1

shall

assume

is

G-) is

G

OC/(

F

G

)

II

f

l\

i)

g

/|

This concludes the proof.

.

||

Whenever

l^^T^V

F e/^V

every pair

ZT!^iO(/( F}

If

=

norm on

be a given

follows then that ot/

Quite often

T2%T3^

for

II

Oi&vooQ

C^(2mi,F.(i G.)^?

*^<2^*^

be fixed.

0(,'(

is finite for

)

=i.2

we have,

g

This clearly implies

G

(|

G

and

<.

)F(f)G(g)|

II

of expressions

Therefore, the set of elements for

in

For any crossnorm

Let

Proof.

dense

is

Then the set

.

1.2

ou7i^V

**c*i

which these expressions stand

LEMMA

,i=

tt

*i

*

denote two sequences of

,

elements dense in '^ and O^* respectively

2,

29

G ^^^C

then

also finite for every expression in

also a norm.

that the

crossnorm

06^

is

uniform, that

satisfies the following condition: VI.

cUZ^Sf^Tg.)

for every pair of operators

S

<.

Ill

and

The geometric significance for

and

S

T

as above,

we

Sill

T

III

etCE,fc8c)

Till

on

and Tj^respectively.

TJ^

of this condition is clear.

define an operator

follows: (

S

T )(2^,fc

g.)

=

2l7.,Sft

T 8i

.

S

9 T

Furthermore, on

TJl

&Q^, as

is,

.

30

This operator

mean hereby

(Lemma if,

CROSSNORMS

II.

2^

if,

f

=

Till

T

S HI

sfll

invariant under equivalence.

is

it

k.

2L..e h-

the operator |||s

We

^

g^

is readily seen that a

It

1.6).

and only

uniquely defined, since

is

implies Sl^Sf^

crossnorm

on

on V^

^1^

uniform

is

T-^

Tk^.

satisfies the condition

*P;^J^

tf/T|/|

OL*

Tgc/^^J^i

.

crossnorms are those which

shall see later that the interesting

satisfy condition VI.

3.

The bound as a crossnorm.

DEFINITION

where

is

sup

^G

and

For

2.4.

lEffc

taken over the set of

"^(2^,^ from T^

LEMMA

2.6.

8/^

into

2Ci F fJ G(g^ (

is

*)^

=

or

2T^,F(fJg^ g

~ *^

=

d

satisfies condition

That

define

numbers obtained when

all

J=

F

the expression

2E^

f f

c

a uniform crossnorm.

a11

may

> (JS^ fj t

Ffi."^

for all

as

represents the bound of the operator

l^determined by

Proof. Definition 2.4 gives

2l7I|f

we

1

^

if,

T?

varies in T ^, and Irrespectively.

Clearly,

only

7

in

g^

F

G^

e^*

gj

^9?-

=

if

The last happens

and therefore

be concluded from

and only

Lemma

II

and

III is

if

and only

1.8.

Thus,

I.

satisfies conditions

if,

if

immediate.

We

and

if,

9^

CROSSNORMS

II.

Condition IV

We

Lemma

a consequence of

is

31

1.4.

shall check condition V:

>(f

=

g)

(8)1

-UPJJ

JJ

G|[

F(f)|

I

" ffl

Finally

we

shall prove that

^

represent operators onl^, and Irrespectively. tflS*U|

llFll

OT*(G)H|||T*|I|

||G||

=

IIISHI

=

III

Let

S

Their adjoints

S

satisfies condition VI.

denote two operators on ^, and Irrespectively.

|fsV)/l

" 8 "

and

T T

Clearly,

and

llFll

llGll

Til/

and

.

Definition 2.2 gives |(

^ The

F 9 G )CE,S^ >'(

SV

1*G

last -- since

A

Tg

= l(

)l

> (SE^fc

)

is a

S*F

T*G

<

)(2L7;,f c

g 4 )l

%)

crossnorm (as

is

proven

in

Lemma

2.7

which follows)

-- is clearly = ||S*F|| |/T*G!|

^

/|F

I)

II

ell

Ills

III

xsr^f.ag,,) Il/Tlll

^(St.f^g,)

Since our inequality holds for any pair

'MXT-.sf.

Tgj^iilsHi

F

ii/Tiii

,

.

G

,

Definition 2.4 gives

xzrs.^Bi)

This concludes the proof.

LEMMA Proof.

2.7.

Let

F

A

is a

and

G

4MM that for any expression^-C^fj

crossnorm.

be fixed.

It is

g^ we have,

a consequence of Definition 2.4

52

CROSSNORMS

II.

FSOCr-^g^l Thus, Definition 2.2 gives

Lemma

with

2.5

LEMMA

F *G

*X

(

^

)

G

F

II

=

)

II

G

If

F

(J

G

II

II

G-

.

tf

This together

.

.

By Remark

Consequently,

2.1,

^

is a

This concludes the proof.

2.1.

The associate

2.8.

Fll

/I

every expression ^E-TT.F. 9*4

Lemma

crossnorm by

>'(

proves that

is finite for

A

$ IIFII llGfl

with a crossnorm

oc'

^^

O6

is also a

crossnorm.

Lemma

Proof. Ot'(

G

F

)

>'

^1

(

v

/

oC

2.2 gives

G

F

=

II

)

-^

Fl)

'

A II

In particular,

.

G

11

An

.

Lemma

2.7 furnishes

Lemma

application of

2.5

concludes the proof.

THEOREM norm

if

and only

whose associate

The associate OC with a crossnorm OO

2.1. if,

is

Ok

^A

.

Therefore,

cross-

is also a

represents the least crossnorm

A

also a crossnorm.

Proof. Suppose that for a crossnorm

and an expression2I t m| f^

OC.

'

g

.

we have gj <.

oeXaTS^i

By

Definition 2.4, there exists an

OtCZlc^flP gj

By

1]

8J

a

F.Tg?and

G// .<

1(

F

G

'

G^/>Ji*such that

)(2T2^i f

,

Definition 2.2, the right side of the last inequality is

oj(

Ok

IlFl/

^(SlTi fc

is

ciate

G

F n

d

Lemma

t

)

OCdE^, *

a crossnorm.

is also a

2.8.

g c)

.

Thus,

F

o6'(

G

)

8^)1

not greater than

>||F

/(

f|

G

Therefore, whenever for a crossnorm

crossnorm, then OU "^^

This concludes the proof.

.

//

,

that is,

ots its

asso-

The converse was proven

in

CROSSNORMS

II.

REMARK

character"* of the

spaces 7* and

'^

is

sup

crossnorm

A

.

seems proper

We mean

u emphasize the general

to

hereby that for any two Banach

any special restrictions, the crossnorm A

J^v,without

on

lZ^F

F

Let

2.9.

for all

t (f)G L(g)|

f

L

for

,

1

T;

g

,

=

,

V

.....

f

I

,

n

=

.

gll

/)

Then, =

1

equal to

sup

(2)

We remark

functional on a

sup

for

.)|

Proof.

is

this point it

T^ % can be always constructed.

LEMMA (1)

At

2.2.

33

|F

first that

aU

if

represents an additive bounded

I

Banach space T? then (f)

for all

|

f

e.T

,

=

1

jj^ll =

1

jl

f|/

equal to sup

for all

|<(F.)|

^

Both numbers obviously represent fixed

go

e1?^,

sup

||g o l|

=1

,

|j

fe

1

T^

P^

||

*.

Q,

pp. 54-55J

substituting

lZ?s

for

Gc ^) F'^ (

.

.

Therefore for a

for

Fo

we

8 et

f

is equal to

sup |2r

This proves that up

A

equals to

(1)

l2<< Ft) G

;<s>

1

similar reasoning proves that the last number equals to

(2)

.

This con-

cludes the proof.

The preceding

Lemma

permits

to

express

^

in a different

form, for

CROSSNORMS

II.

5^

Banach spaces which are conjugates content of the following

LEMMA (

s\

F-

LEMMA Then

ot/ is

may be

G-)

also a

Proof. That

21J=*t 1^

9

in

Gj

also represented as

a consequence of Definition 2.3 and

is

Let

2. 11.

Banach spaces. This forms the

Lemma:

For an expression

2.10.

Proof. This

of other

O
represent a crossnorm

^"^ on

crossndrm

cst'

is a

crosshorm

Lemma

2.9.

^

on

^ Theorem

is stated in

2.1.

Definition 2.2

gives

sup

= By Lemma

2.10 the last

UP *>^

Hfll Hg

|

number equals

to

>(3EjI,^

II

This concludes

&f)

the proof.

THEOREM

also a crossnorm.

Then, oc"

Proof. Since

Theorem

2.1.

Let dU denote a crossnorm whose associate

2.2.

*6 and

,

ol.'

csC

,

*

......

Lemma

we have Ot

2.11

Thus, by Theorem

norms. This concludes the proof.

is

are also crossnorms.

are crossnorms,

Applying successively

Od/'^>on Tg*OT&*

.....

c*

we

2.1,

get OC/ <**

.

^

A by

^>^on

<*?

......

JjS?<s> Tfi

,

are cross-

CROSSNORMS

II.

Theorem

A represents the least crossnorms whose

2.1 states that

associate is also a crossnorm. sent the least crossnorm.

when in

1J.

35

We remark

In fact,

we

that

"X

does not necessarily repre-

shall prove later (Chapter V,

%

11) that

and 1-^denote two two-dimensional Euclidean spaces a least crossnorm

1^01^^ does

By Theorem

not exist.

norms whose associates are The following interest.

It

between the

Lemma

proves that f^'s

cross-

2.1 this implies the existence of

not crossnorms. 1

concerning the "local character* of

^M^L^,^

g^)

and those between the

"^

is of

depends only on the spatial relations and not on the including them

gjs

spaces.

LEMMA

Let TtL^ and TTL^denote two closed linear manifolds

2.12.

Banach spaces TJ^and "Irrespectively. Then of

>

Let 2T,ft

shall prove our

bers (Definition (a)

on

^^^^

is

an extension

7

on

Proof.

We

")l

in the

sup

>

gc

Lemma,

be an expression in

by showing the equality

7^0771^ C.

^T^

of the following

.

two num-

2.4):

|2,F(f

)G(g < )

|

for

F

&

ftl*,

G

for

F e

7?*,

G e 7^5

6. 7fl

l|

F||

=

||G|)

=

1

=

1

and (b)

sup |5T

We

recall

(

F(f;)G( g

.)|

Hahn-Banach*s extension theorem

an additive and bounded functional

F

[1, p.

I|F 55]

= JJ

G

Ji

which states that

on a linear manifold "W^ in a Banach

space 1? can be extended (quite often in a non-unique manner) to an additive and bounded functional

F

on

TC>

,

without affecting the value of the bound,

.

56

CROSSNORMS

II.

that is, =

F(f)

-

-

sup

(I

From On

this

f

tional on 7^j

,

as

theorem

we may

sup

it is

when

/If

obvious that

F

f

^

(a)

It is

)

(b)

denotes a given additive and bounded func-

restrict ourselves and consider the values of this func-

an additive bounded functional

e TTC,

and

If

tional only on the closed linear manifold T^l^CT^.

f

TfL

<

,.

<Jt-F>t

II

the other hand,

for

F()

clear that

F^

||F||

on T/C^ (such that

^ F ||

||

p,

F

Thus,

p. 54]

.

on

F*(f)

=

Now,

let

^ defines for

FJ,(f)

F

and

G*

denote two non-zero additive bounded functionals on "^, and ^^respectively, while

F

and

respectively.

G^

stand for the corresponding functionals on

last inequality holds for any

(b) .$ (a)

4.

.

II

jz

F^ Tg?

,

^

Fj|

|l

Gj

Ge ^^

.

Therefore,

This concludes the proof.

The greatest crossnorm.

In the

arguments which follow we prove the existence and actually con-

struct the unique greatest crossnorm.

DEFINITION

We

|

Then,

llF^llllGMl

The

and

'Vft

define

2.5.

Letjg^f*

g

be a fixed expression

i

CROSSNORMS

II.

where

taken over the set of

is

inf

toz^.rtsg'

all

37

expressions

8 2?^ v O

equivalent

g

^

.

LEMMA

2.13.

is a

^

uniform crossnorm, that

is,

^

satisfies con-

ditions I--VI.

Proof.

I:

Let2L* (

with

F

II

l\

By Lemma

=

we

1

IJi*

.

have,

extreme

1.2 the

F

For an

denote a fixed expression.

gc

f,

left of the last inequality is

invariant under equiva-

Thus, Definition 2.5 gives

lence. II

Zl7*F(fJg

Now suppose

F

pose that

=

II is

^ ^(Zr=

F|l

g

1

.

gj

is a

and

such that

Similarly we can find

FC^with

equivalent to

Z^^tfjg^ >

consequence

2^, h c k^

& <- Z^

f

c

*

(2L^i f i

.

.

gj

F)|

By Lemma

Naturally

8c

=

1

.

1.7

we may sup-

>

of Definition 2.5 for

^

.

be two given expressions and

find an expression

2::.,

such that

which

for all

g.)

t

Consequently,

immediate and IV

we can

,

is not

.

C "^*"for

Let 2T7^, f c

Ill:

Clearly,

||

that JSE-jL, f^

there exists an l|

-

%>

>Q

58

CROSSNORMS

II.

We

have,

Condition IV and Definition 2.4 give,

The

^

last inequality holds for any

V: -

F(f)

Let

llf

:,

and

II

f

gc

c

llF

C^

=

II

f

1

g

We

.

p. 55]

["l,

This proves

.

choose an

By Lemma

.

III.

F

l^ such

that

1.2,

Consequently,

Thus, Definition 2.5 furnishes VI:

Let

T

and

S

^

(

f

8

= )

H fH K g

//

denote two operators on T& and 7i

respectively.

Then,

III

HIT

sill

By Lemma

1

.6,

and therefore while the

111

(aS.flfJ llsJD

Z7X,

g^2Tp, h^*

f^CB

Tgc

-^ (Z'ct.Sf.fll

we are running through

extreme

= )

k

'

a

implies

T^ 2-^,

^T

Sh>

Tk ')

This concludes the proof.

is the "jj

follows that

remains invariant. Consequently,

Definition 2.5 gives

2.3.

Jt

the set of all expressions equivalent to2-,f.fl

left of the last inequality

THEOREM

'

greatest crossnorm.

CROSSNORMS

II.

That

Proof.

crossnorm was proved

is a "Jj

o(/

39

denote any crossnorm and

Z^,

Lemma.

in

2.13.

Now,

let

Then for any

fj* g^ a fixed expression.

expression

>;-^, cc

**>

f

f

we have

Thus, Definition 2.5 gives,

This concludes the proof.

REMARK character*

We

2.3.

notice that

that is, for any two given

1

,

^

(as in the case of

"y

Banach spaces

*)^

is of

)

"general

and T^^without any

special restrictions the unique greatest crossnorm can be always constructed. In general,

however,

analogous to that of

we

1

not of "local character*

is

-^

Lemma

2.12 (proven for "^

^

is

is not true.

LEMMA b

x

,

2.14.

bj^

,

Let

a.,

n

denote

1j^

,

lemma

for 'Jj

Later however,

"Y-

,

for

which

preserved.

For our further discussion we

,

that is, a

shall point out the precise conditions on the spaces

the local character of

b

)

,

a

,

shall need the following simple proposition:

,

a^^

denote

positive numbers.

n

real

numbers and

Then,

max b +

Proof.

Lemma

The proof can be carried out easily by induction, verifying our

first for

n

=

2

.

*K>

CROSSNORMS

II.

THEOREM

Let

2.4.

f

be a functional of expressions on

1j

satisfying the following conditions:

assumes

For equivalent expressions O

(i).

\&(*Z

(ii).

6

Then, "

*^* tf2, Proof. Since

<

(

*,<*

f

~

.

sup

gj

g

*;

=

f

II

)

|) ||

g

ff

f

3

g

(Lemma

2.13, V) the right

side of our equality is clearly not greater then the left side.

.2^^ (i)

and

be fixed.

g^ (ii)

for *3

By Lemma

< ^

Then

for any

---

extreme right

I^Kg.-)l

max

expressionZlT^

ff

Now

let

C^^T_

gf

f.

g

furnish:

2.14, the

'*^**

value.

t

t

sup

same

the

is

.

^

^,

sup

f,^

iifjiii/gjii

This by Definition 2.5 implies

II

The last inequality holds. for every expressionS^ifj the left side of the equality of our

theorem

is

g^

.

f

II

f

a

II

g

g II

Therefore, also

not greater than the right side.

This concludes the proof.

THEOREM tion 2.5) on

^

|

2.5.

^>

The associate with is

>

the greatest

(Definition 2.4) on

crossnorm V (DefiniTO

,

that is,

'^^

.

CROSSNORMS

II.

Proof. For a fixed expression^E.^m| F^

# G^

obviously represents a functional of expressions satisfying the condition of

Theorem

By

2.4.

Tq'^,

in

^L^

f

J

g

.

the

number

in

Definition 2.2,

-.up V'CSTT ^ ^'OF..G.) " *=.^-

By Theorem

The

last

2.4, the right side equals

number represents

concludes the proof.

^

(2TT* J*

F 1

*

^

G) o

by L,emma 2.10.

This

CHAPTER

III

CROSS-SPACES OF OPERATORS *

1

The Banach spaces

.

For a given norm in general, will not be

We

by adding new elements. is,

In that case

complete.

into the smallest possible

it

the

crossnorm) o

(or

Banach space

*B

linear space

we "complete"

it

that is,

&&%

imbed

usual Cantor-Meray fashion

in the

consider namely

normed

all

fundamental sequences (that

those which satisfy Cauchy's condition) of elements (or expressions repre-

senting those elements) in identifications [5, p. 106}

A

(i)

7?J

^L.^*. and introduce the following standard

:

sequence consisting

of identical

elements we identify with that

element.

Two fundamental sequences

(ii)

and only

if,

(iii)

limit of the

the

norm

The norm

norms

space *^i by a

Tj,

*Q?,

^LL'?>

fundamental sequence

of a

elements

3.1.

in that

anc*

elements

of

a**direct

*p <8^ fj^ is (

we ODta ^ n

*ty

of the

c*,

product* of T> and '^>

termed a "cross-space*

^U'^

1

anal

term

is

defined as the

sequence.

The Cantor-Meray completion

termed

if

toward

^7^. which naturally depends on the norm

crossnorm, *9

elements we consider identical

of their difference tends

of the

DEFINITION

of

it

.

,

.

normed

linear

will be denoted

Whenever

oC is

Similarly, completing

the space associated with

CROSS-SPACES OF OPERATORS

III.

^ (8.1^

or the associate space for the cross-space

REMARK This

finite.

always the case whenever,

is

is

.

defined whenever otx is always

ot

The inclusion

2.

Let

Theorem spaces

The associate space

3.1.

l

denote a crossnorm

cf*

,

^Ok-

T^j

T?^, and a given crossnorm

ol

1

expression

Thus, two given Banach

^-^ deter mine the spaces

l^X.Tg.

2^ R- &

G.-

the "inner product*

sents an additive functional on

Since by definition

f^j

^v

1j

is

O^^

dense in

Theorem

(2^,

fecting the value of its bound.

REMARK

defined in

f\

9

The

3.2.

7?\

U,'^?V

p. 180J

explained above.

,

We may

with a bound equal to

O(^

"^^otTra^ ,this functional

T^

<S^

repre-

8^)

(5y,

1^-

can be ex-

vL

without af-

more than merely

a certain subspace of

1

(

^ot*^,

)

the

as

since that equivalence is to be understood in the sense

Throughout the rest

of this

paper

all

inclusions of this form

will have to be understood in the sense explained above.

It

Q.)

write therefore,

last inclusion represents

w * tl1

2.1 that for a fixed

Q)f^r/ V*

5'

tended in a unique manner to an additive functional on

equivalence of

By

(Definition 2.4).

.

1J^

a consequence of Definition 2.2 and

It is

^^

*p|

also a crossnorm on

O(/ is

2.1,

^

^-^ on

does not appear to be a simple task to state the precise conditions

imposed upon a crossnorm ok

for

which the resulting cross-space

kk

is

CROSS-SPACES OF OPERATORS

III.

such that

conjugate space coincides with

its

always the case, for instance, when finite

(

7?;

We

dimensional.

^L^

^ ,A

all)

operators from

TJ* into

ator is in general different

T^

*^,

(from

from

its

^L

(Definition 2.5 and

3.

VH

(

fljpt

DEFINITION oC-norm,

An

3.2.

(

A

for

II

A

of an

oper-

the greatest cross-

2.3).

Chapter

A

operator

g^.

in

and term the

Ij^

which such a

The

(not necessarily

where the norm

),

when we deal with

o(/-norm.

06 stand for a crossnorm on

let

from 7^

there exists a finite constant

if

expressions^T^ fj&

denote by

Theorem

of this

or Tj^is

although this is not stated each time explicitly.

7^^

for all

1^>.

is

by interpreting

some

of

as the space of operators of finite

*?Q

Throughout the rest *pj

into

T^,

This

bound. Furthermore, we are able to pre-

sent a complete discussion for the case

norm Y

this direction

Banach spaces

as

ot'*?^

associate space.

one of the spaces

present the first step in

as well as

)

at least

its

*^O **

finite constant

^p!^*

O

into

of finite

such that

The least

o(/-norm*

Ti*is termed

of

A

of such constants .

For an operator

does not exist, we define

justification of such a definition will be clear

||

A

{/

.

s=r "t>

from Theorem

3.1

which follows.

LEMMA linear space

Proof.

if

The operators

3.1.

jl

A

Jl^

A

represents the

of finite

norm

The proof does not present any

of

-norm form a normed

c

A

difficulty.

we

.

CROSS-SPACES OF OPERATORS

III.

LEMMA

A

II

for any

crossnorm

I

Thus,

fl|

A

HI

(Af

^ A II

THEOREM space of

all

A

Tfr

^

into

we have always

III

II

A

Definition 3.2 gives

,

(f0

11^

*Q&Jfe

(

A

from

A

of

g

,

=

g)

II

AH,llfll/|g/l.

This concludes the proof.

.

J^

3.1.

An element

f

*?

)g|

norm

III

from

.

operators

represents the

TjJ>

OL

5*

I!*

For every pair

Proof.

A

For an operator

3.2.

1^

.In

into

(

rnay be interpreted as the Banach

<^,-norm, where

of finite

7^>

generates an operator

)

A

oL-norm and conversely. This correspondence

into ")^*of finite

II

A

//

other words:

^ ^^7^

(

JT of

TJ

)**

from is

such

that:

(1)

V

<""*

A

(2)

1*

A

(3)

IIJiTlll

Proof.

for any scalar

and 9^ ^r?

f

A

II

Let9*

a

implies

]j

A^

^

aA

implies

whenever

3

^r

-

H*

9^ ^r?

*Jj

^< ^^ )

***

and

9*111

II'

denote

its

bound.

In particular,

_,

g ,

-f

A

f

A

a

y

A

ot

7?>

(

1

^

=

&(t

)

\\

_

Ci

/

) t

*

*j (^,

g)

Clearly,

k6

CROSS-SPACES OF OPERATORS

111.

Relations

and

(i)

prove that for a fixed

(ii")

sents an additive functional of

<<

I

Therefore,

f

8

o

y i

Af

a

A

'

OU

8

=

(

)

II

J^g g

)

repre

bounded since

is

f.ll

)

II

g

unique element of

6i J^, a

f

III^H!

(

for

II

g

1^ which we

)

for

g

f

^,,

g

(ii)

is additive.

Since 7^ is the

Af

(

)

That functional

.

^

,

so that

,

"=

g

and therefore by

Thus,

Ill^lll

assigns to every

shall denote by

<&(

<

) I

g

^

*pu is

C&

same, and thus

dense in

Q &^

r

,

Tp>

the bound of ^7

represents the least constant

y'l

on both spaces

C

satisfying the

inequality:

for all expressions

By

Xcl

g^

f^
in

1(^

07^.^

.

Definition 3.2 however, that least constant

denoted by

|1

A

bounded since

11^

111

A

Therefore,

.

^

III

Thus, an ty in 7^, into

t

iaof

finite

(

06

II

A H^

1^

^oilr^

-norm

since by

Lemma

again by

(b)

-^

1.5,

above,

\\\<$

for which

A

f(

A

from

of (a) above.

is

(

\\\

A

<+oo.

was is

Af^Jg,;

A

determines a unique operator

'

fy

=

TJ> into is

is invariant

bounded on *R

from

||^

4^,

2.^, *Q

= I|^

satisfying (b)

.

Conversely. For an operator

we construct Of by means

A

II

C

^tc

T^ with

/I

A

Jl^

uniquely defined on

< -f

T^ G

under equivalence. Then, ^ ence on

'^rj

Li^-x an ^

CROSS-SPACES OF OPERATORS

III.

= jj

A

f

The correspondence

.

.

(|

A

1*7

obviously additive.

is

This concludes the proof.

COROLLARY is

complete

Proof.

norm

in the

The proof

The space

4.

LEMMA Moreover,

Proof.

equivalence

3.3.

|JA

The space

3.1.

||

A

//

of all

.

3.1.

A

from t^

into

is of finite

7^

"J

-norm.

.

III

A

For an operator

(Lemma

Theorem

operators.

Every operator =||/A

OC-norm

of finite

jj

a consequence of

is

A

of all operators

Consequently,

1.5).

^

/

the

,

sum"2l |

2T^

(Af.)g^

is invariant

*s

(^^.)8^l

under

a functional of

expressions satisfying the assumptions of Theorem 2.4. By that theorem,

II

A

|L Q

=

=r

sup *Si\&fo

sup

Y(^^ i8-)

^>5

f

The extreme right clearly represents the bound

of

A

H

M

I

8

I

This concludes the

.

proof.

THEOREM space of

all

3.2.

operators

(

f

may

7^ Cfyy )*

rom 1^

into 7i

,

where

be interpreted as the Banach the

norm

of

an operator equals

to its bound.

Proof.

The proof

is

a consequence of

Theorem

3.1

and

Lemma

3.3.

'

A

5.

"natural equivalence*.

REMARK fy for

(

f <8 g

which

CROSS-SPACES OF OPERATORS

III.

1*8

^e

Let

3.3.

(

^^T^)*

A 11^

||

= III;?

A

ing" operator

from

Ill

For

a fixed

additive and bounded functional on

A fromt^into 4(

That

||

f

g

= )

= Ill^lil

3U|^

in the proof of

Theorem

is

=

ipj

same fy

shown

oC-norm

finite

=

III

11^

'J[

,

^y

(

(

$fll

f

g

f

g

This can be

.

represents an

)

also determines

)

for

<,

f

manner analogous

in a

g

,

to

^^ v A

fl

//^

= HI^I/I

3.1.

in addition to

)

^/ft

(g)

For "corresponding* operators the

A

U

Thus,

.

Tjof

into

however, determines a "correspond-

which

g

^

3.1,

Furthermore,

.

^j

(A)g ,

from

The same y

.

7^ into 1%P for

readily seen as follows:

a unique

A

determines an operator

)

Theorem

Defition 3.2 and

By

.

A Jl^

!{

oC-norm

of finite

=

If

X

we

]f^

(that is, generated by

also have

111

A

=

Iff

)/|

A

|/|

.

This can be seen as follows: III

A

= III

sup

II

Af

||

=

sup

14

sup

Hfli*l

(

f

g

)|

=

1*11*11311*1

||Xg||

=

III

A

.

IK

li|iii

Thus, if

in their

all

preceding

Lemmas

and Theorems of this section are also valid

wording we replace the phrase

"operator from

t^

intol^^.

**

operator from 7^ into f^,* by 1

In particular

as the Banach space of all operators

general the corresponding operators traction of the adjoint of the other.

(

^

^L^X)

fromT^into'^

A

,

A

,

of finite

is

expressed in detail

cC-norm.

are such that one

For reflexive Banach spaces

ing operators are adjoint to each other.

This

rnay be regarded

in the following

theorem:

is

In

a con-

the correspond-

THEOREM Banach space

III.

CROSS-SPACES OF OPERATORS

3.3.

There exists a "natural equivalence

A

fj of all operators

from 1^

A

and the Banach space 1J% of a ^ operators

o-norm

sense that

in the

A

axA %

(i)

a

(ii)

A* r>

+ f

f

(iii)

flAll*

(iv)

Iff

= I!)

The proof

Proof.

6.

A

_

LEMMA

3.4.

rank are of finite t^ into t

^

^

a

;A

For

IIXII^

between the o(,-norm

into *^a of finite

from l^into '^

of finite

|

a^X^

is a

(

*

D

a

>

A

are real numbers).

a^t

,

.

.

lUXfll

.

consequence

Remark

of

3.1.

a space of operators.

a given

o(/-norm.

of finite

+ ;

and

<8d
T^!

"

jp

^

A =

If9

crossnorm

OL^*X,

the operators of finite

The normed linear space

rank^where

||

A

||

of

operators

represents the norm,

may

A

from

be

characterized as

Proof.

A

F. given expression 2^T O <) '

indicated in the proof of

defined by the relation

from *^

into

spondence finite

is

*^

Theorem

3.1 to

Af =2! o

G- corresponds <) an operator

F;(f)G a o

-

. '

.

A

in the

from T^

Since

clearly additive.

O6

^>

the value

,

ot'flSlT*,

by Theorem 2.1. Since,

Fd 8<

V (S -.

Definitions 2.2 and 3.2 furnish

f

cSc)

=

into

*^x

Conversely, every operator

can be obtained in such a manner.

of finite rank

<*?=,

manner

2-7./ Afc)c

The corre-

K-8

Cy)

is

50

CROSS-SPACKS OF OPERATORS

III.

=

,'(2, F.9C.)

llAll,,.

This concludes the proof. 4

THEOREM may

3.4.

be interpreted as the Banach space of

oO-norm

of finite

(with

be approximated in that

A

Proof.

^

Let ot be a given crossnorm

))

A

11^

norm

all

"X

norm

Then,

A

operators

representing the

.

of

Trf^.^x

from

A

),

Tr^ into

T^

which may-

by operators of finite rank.

given element in ')^

*^v is

^

represented by a sequence of

expressions

in

^PG^

fundamental relative

*?,

norms

to the limit of the

senting

quence

it.

By Lemma of

Ap

of the

Ap

3.1, the

lim

p -*o*

3.2,

the given element.

norm

of

such an element equals

expressions in the fundamental sequence repre

II

A IL~

AP

A

A

intoTJ^for which

.

space of operators of

II '

T?>

1>

A

determines an operator

By Lemma

the

operators of finite rank from

lim

Hence

;

3.4 such a sequence of expressions determines a se-

p,
By Corollary

o
to

-

*

A JljBs *

finite

of finite

06 o(,

-norm

-norm

is

for

complete.

which

.

I*

is

uniquely determined;

it

may

be identified with

Furthermore,

This concludes the proof.

THEOREM

3.5.

^

=

fc

represents the Banach space of

operators from*^ into'^> approximable in bound by operators of

finite rank.

CROSS-SPACES OF OPERATORS

III.

That

Proof.

quence of

Lemma

THEOREM

T^ of

sr"Xwas proven in

3.3 and

3.6.

=

)* into

"ft

Theorem

Theorem

is a

conse-

3.4.

occurs

"^TV

and only

if

ot-norm can be approximated

finite

The rest

2.5.

Let oC denote a crossnorm

^^d'^x

51

The equality

.

every operator from

if,

in that

norm by operators

%*.

from

Tfe into

T^

The proof

Proof.

LEMMA ?au

of finite rank.

is a

consequence

For any crossnorm

3.5.

are completely continuous

Let

Proof.

A

p,

lim

II

A

is

A,,!!!^

lim

completely continuous by

REMARK the

A - Ajl. " ^

.

into

Ap

3.4.

1J3

deter mined by 7

of finite =

rank we have,

.

3.2,

jlim^lllA

Thus,

Tfi

and Theo -em

3.1

the operators determined by

p. 96^]

be an operator from

p-T-oo

Theorem

oC *^*^

Thus, for some sequence of operators

By Lemma

of

3.4.

Banach space

An example

of all

j|

A -

fl, p. 96,

for which

*^

=

A^^

.

Theorem 2J.

<8f~"^

is a

proper subspace

of

completely continuous operators from T^ into 1^,

would answer negatively the "basis problem"

i, p.

Ill]

.

Suppose

A

is

a completely continuous operator which cannot be approximated in bound by

operators of

finite rank.

separable Banach space

The range

'^ Q,

of

p. 96,

A may

be considered contained in a

Theorem l]

.

It is

well known that a

completely continuous operator on a Banach space whose values

lie in

a

52

CROSS-SPACES OF OPERATORS

III.

Banach space having a

basis, can be always approximated in bound by operators

*

of finite

rank

approximable

It

[l lj

in

bound by operators

THEOREM

3.7.

Let

proper sub space

of

Proof. The proof

At

this point

Otherwise,

A

is

of finite rank.

cL denote a

operator of finite 06 -norm which is a

^ must not have a basis.

follows that

crpssnorm ^*X

.

If

there exists an

not completely continuous, then

is

(

is

a consequence of

we are ready

Lemma

3.5

and Theorem 3.1.

supplement Theorem 3.3 with the following

to

assertion:

THEOREM 1

lence* of

3.8.

Theorem

Let

3\3 is

between the Banach space

which are approximable

Banach space

of all

approximated

in that

Proof.

oC,

be a crossnorm *^~X

such that

norm by operators

operators fromT^into *Yp?oi

norm by operators

An expression 2^, F? 9

sponding" operators 2ETiF.(f)Go d

operators equals to oc'(S^ ^

arguments

in the proof of

Theorem

t

G.*

F.

o

3.4.

finite

of finite rank,

finite

o^-norm

and the

oC-norm which may be

of finite rank.

in

'*<& T^ determines "corre-

^^.G.CgjFa 6

,

*

of these

also preserves a natural equivalence

operators from *Q into Tj^ of

of all

in that

it

The "natural equiva-

.

G.)
.

.

The

<xV-norm

of both

The rest follows from the

CROSS-SPACES OF OPERATORS

III.

The local character

7.

53

y as a characteristic property of unitary

of

spaces.

We

conclude this chapter by settling the "extension problem" for the

greatest crossnorm

Theorem

An

^

,

indicating at the

P

operator if

and only

on a perfectly general Banach space

P

if,

linear manifold of all the

projection

^

be

may

is that of a

for

f *s

1

.

P

=

The range

which

=

Pf

complementary

to *Y/

sented in the form

f

A

.

if

=

f'

+

P

complementary manifold

Let tively.

Obviously than

=

Pf "fft

Let

closed

is the

The bound

f

closed linear manifold

and only

existence of a projection

which

P

of

such a

Closely connected with the notion of a projection

f"

if,

,

every

where

of

7^ on

in *)* is said to

~/

f'eTT^ and

f"e It

Lemma

TjJ*,

readily

that the

"flL is equivalent to the existence in

t to T7L (namely,

T(, is the set of all

f

*s

).

>

gc,

be a fixed expression in

By

Tfc

and

ift-^ is

expression of the greatest crossnorm on

^?*3u

respec-

equivalent to

Definition 2.5 therefore, the value for

crossnorm on T7c.0

/

Tl^oT/t^C

O "^^ possesses more expressions

'V/1,07/5^ does.

of the greatest

It is

.

1.1. 1J

and T^C-jdenote two closed linear manifolds in

2^,fc

be

TJican be uniquely repre-

in

f

seen, as is also shown for instance, in [?, p. 138,

for

of

termed a

is

complementary manifold. Let "t denote a closed linear manifold

Banach space 1^

of a

application of

3.2.

projection

in a

same time an

'

not smaller than the value for that T*> (

O ^^

.

54

CROSS-SPACES OF OPERATORS

III.

LEMMA manifold in (that is,

Let

3.6.

** such that

T/=

Ttt,

7u is the conjugate space of

Suppose further

.

)

denote a Banach space and *WT a closed linear

"Y^

that,

TTU,. C T^'W, crossnorm on ^^
1

of all

bound on

on

~YlC

Similarly,

.

^

yft&fTTtLe

By assumption

into

=

T^ 9*YfC

(

into T^1

such

*y/
TW^T/T, where

from T/C

P^

(**g)

(1)

(

*Tt

operators from

ty>

3.2,

from

identity operator tional

~ *Wl/

into

//

irL>

with

(3)

<(*8)

(4)

IR^HI

Again this ty (5)

(6)

)

represents the Banach space

of an operator is given

and 3.2, the

3.1

(^f)g

^r

feTTL,

TltflL^ ,

is

geTf,

a subspace of

p, 55,

Theorem on

*y

and

2j

,

Thus,

"^o

rfc G$L

y" can be ex-

*Q &>uLc without changing

which

(fg)

=

=|||il||

=

^r

ffcHT,

8

eTTLo

,

and

.

generates an operator

<(fg)

norm

that,

the space

is, for

the

generates an additive bounded func-

tended to an additive and bounded functional bound, that

represents the Banach space

)

By Theorems

.

C

by Hahn-Banach*s extension theorem ^ 1

its

an "extension" of the greatest

Then, there exists a projection of Jj^on

By Theorem

of all operators its

is

.

.

Proof.

by

^^1^

crossnorm on

that is, the greatest

some Banach space

(Pf)g

for

P

fromT^

f^,

g

into

*W^=

e,

and

^L

,

for

which

CROSS-SPACES OF OPERATORS

III.

Clearly,

P

of

is

111

"^

P

=

and

is

on

^

"t

on

7fL/

Banach space

Proof.

,

OC

elements

complementary. Let

is

prove

this, put

=

f. ci

j

=

1,2

f

in

2E^ ^

an expression in

which

,

.....

domain

,

m

f.'

P

1"

equals

P

=

1

.

Furthermore,

P

holds, that is,

i& on "flftwith bound

"Y^t.Q

-f

C

prove that ~tft(E^(X

1

is

implies that

1.

We

denote by

T^O^OC

TC

=

for which

8

^ e a fixed expression in T)fo

TL0 CtOC

f?

f'

4

Pf. A

.7/T,

P

Let

be

/, are

UL

,

and

Then, SLjt'jPf.ft g.

it.

equivalent to =

Tf/C and

.

equivalent to

is also

where

Pf

.

the closed linear

T^

<J

.

P

of

.

any expression in

]Jg^f.<8J g,

the

denote a closed linear manifold in a Banach space

is sufficient to

It

all

\

.

the relation

a projection of 1^* on I^Cwith bound

manifold of

f^

This concludes the proof.

.

Let 7f

3.7.

i

Then, the existence of a projection of

for any

an extension of

Thus the bound

(6).

is identical

a projection of

=

yjp.iii

(2), (4)

LEMMA

P

prove that

(5)

Furthermore,

.

pin

by virtue of since

and

(1), (3)

55

ST^fT f/fe"fC *

g

.

To

for

Thus,

and therefore,

The

left side clearly,

represents an expression in

side is an expression in

TfcdC

"() OC

,

while the right

Thus, both sides must be equivalent

to

ft

CROSS-SPACES OF OPERATORS

III.

56

& o Therefore,> 3L7>,f.^ fcl v

Since

Thus, the

*ft@OC

P

inf

-

o

^_,

i,

equivalent to^ET^fTlS g?

7/

with bound

we have,

1,

expressions from

not smaller than the one obtained by

equivalent to 2El,f*(8J g*

On

.

the other

This was pointed out in an argument pre-

This concludes the proof.

3.6.

COROLLARY

is

,

from VfCoOt,

hand the converse always holds.

Lemma

^^ on

in Definition 2.5 obtained by taking all

taking all expressions

ceding

^

denotes a projection of

3.2.

Let T/t denote a closed linear manifold

in a

Banach

*

space

.

space TfL9 for any

.

Suppose further that

7f

is the

Then, the inclusion

7H

>Tft

Banach space

Proof.

The proof

COROLLARY space ^/

.

Proof.

Q~

3.3.

conjugate space of another Banach

C

T

/>ff

implies 7

y

CC C T^G^OC

.

is a

consequence

of

Lemmas

3.6

and 3.7.

Let 1ft denote a closed linear manifold

Then, "tfC(9.0t

C

OL for any Banach space OL

fv GL

For a closed linear manifold TfL

projection of Fv on T*L/with bound

in a Hilbert

An

1.

in

.

Pv there always exists a

application of Corollary 3.2 concludes

the proof.

COROLLARY in the

3.4.

Banach spaces

a projection of *y on

Tp,

LetT/^and

T/tjdenote two closed linear manifolds

and T^respectively. Suppose further

"t, with

bound

1

and a projection of

y >.< k

th'

t

there exists

CROSS-SPACES OF OPERATORS

III.

bound

Then,

1.

Proof. Applying twice Corollary 3.2,

and

T^LT7tx C lSL^?a.

THEOREM

A Banach

3.9.

space

c

7^

II

We

get

TJl^Tfl^

C

recall that a

is

unitary

if

and only

if,

-g^Ttt*

for any two-dimensional linear manifold

Proof.

we

This concludes the proof.

T/t^-m,*

norm

57

fft, C.

Banach space

Tp

TJi is

.

unitary

if

and only

if,

its

satisfies the relationship

f B

+

for every pair

The proof

and

i f

is

f^ in

lp

.

thus a consequence of Kakutani's characterization of

unitary spaces which states that a Banach space is unitary exist projection operators of bound

^

,

and of

Lemmas

COROLLARY

3.5.

1

if

and only

if,

there

on every two-dimensional linear manifold

3.6 and 3.7.

Whenever a Banach space T^

exists a two-dimensional linear manifold

is not

unitary there

^u^7?for which I^Cklfu

is not

an extension of

Proof. The proof is a consequence of Theorem 3.9. It is

not without interest to conclude this discussion with a stronger

statement than Corollary 3.4.

58

CROSS-SPACES OF OPERATORS

III.

THEOREM

Let

3.10.

Tfifj

and T/Zjdenote two closed linear manifolds in

P

two Banach spaces T^ and Irrespectively. Then, there exists a projection of 1^,

onTTtjWith bound

and only

^^on

7/t^with bound

1, if

and

VflftTfCi

(2)

there exists a projection

Suppose that

To prove

(2)

we

P

1.6).

range

is

|

a,

of

*9 >,* r7?L on

exist.

I^_

^A-^^a.

with bound

Corollary 3.4 tells us that

1.

(1)

put

P

Clearly,

TfC O')Y

P

and

f

P

not difficult to see that

(Lemma its

C #,V2v

(1)

holds.

F^ of

if,

Proof.

It is

and a projection

1,

uniquely defined on

is additive

Now

.

is

on

T^

1^

^>

identical on

,

7

for any

g

f'.

expressionST^

CZ-JE^ff

g^

we

have,

Thus, Definition 2.5 gives

Thus, the bound

P

of

on 1^ '

manner

an operator

to

on

P

O 7^ "o

on

1^

Conversely, we assume that that

G

)(

g

II

=

<S^.

Vft'rtl. with bound

1

.

By

on Tfl % such that

(jl,

p. 55J

G(%J

=

II

P

is 1.

can be extended in a unique

P

Clearly,

7^^^.

is

a projection of

1.

and

(1)

(2) hold.

Choose a

g

there exists an additive and bounded functional g

H

=

1

,

III

G

= l/i

1

.

Clearly,

G(g)g o is

CROSS-SPACES OF OPERATORS

III.

a projection of bound

^i

80

We

=

gj

27-, Q

readily verify that

identical values on

bound on jection is a

of 7/t^on the linear set of multiples of

1

^ ^^^ts^Y Lemma 3.7.

Q(Z. f*

*YTL

from

Q

1

go anc*

Tflj fSL^yfL^

projection with bound

1

it

on of

f

g<>

may with

may

cT^

f

or

and

f

Ti and

Tft^.

true.

its

without changing

^ TH

of

L ,

f

on

go

Its

.

bound.

Thus,

QP

"contraction" clearly

W'flL go

.

The

last two

since their elements are of the form

f

(

g

its

g

=

II

)

^onTfl^

.

f

A

II

H g

= l|

II

f

II

.

Thus,

similar statement applies

This concludes the proof.

By Theorem a statement for

1

define

t

be extended in a unique manner to a pro-

^

7ftj,

there exists a projection with bound to

of

T/

71^0

Furthermore,

^Z-^L g

l

be identified with T, and f

is

^T^^ on Ht ^L 1

in

g^

f^fc

invariant under equivalence, assumes

*W.<SL g d

T^j

|

Hence,

.

g^

.

^s range

furnishes a projection with bound

spaces

g

G(g.)f.

Thus,

.

Now, for 2I^

is additive,

*^y

O.'Xrt^is

59

v

3.9, in general,

analogous

to

^jj

is

Lemma

not of "local character". Z.I 2

proven for

^

,

is in

Therefore, general not

60

CHAPTER

IV

IDEALS OF OPERATORS.

Ideals of operators.

1.

For a crossnorm

,

the

Banach space

of all

A

operators

from

7$>

^.

^>

1^ (from

into

OC>

T^intol^

)

ot-norm (where

of finite

A

II

||^

represents the

*

norm we

A

of

be characterized as

may

)

shall characterize all

We

an "ideal" property.

here

in the

We

language of Banach spaces

X*

on

X

f

X

some operator on

X

=

^

,

,

,

ment

1^

,

that

lemmas

is

is

= |||

namely

.

always an extension

X

and every operator

X

and T^T are reflexive

listed

7^

x)||

= (

shall

X*

)

^^^

possesses

)

[JcfJ

.

assume

It is

on

T^

determines an *!**

Similarly,

X

of

^

.

When

*^L

is the adjoint of

.

under consideration are reflexive. This *^j

(

equivalent to "unitary invariance"

|flx*|||

To simplify our discussion we *&

which

on a Banach space

T^ with X**

determined on T^**;

is reflexive

06 for

present section

In the

.

)

to Hilbert spaces.

recall that an operator

"adjoint" operator

T^Ljg^)^

shall see later that the characterization presented

crossnorm when applied

of a

is

crossnorms

(

that the is

Banach spaces

equivalent to the state-

readily seen that

some

below also hold for perfectly general Banach spaces.

of the

IDEALS OF OPERATORS

IV.

A Banach space^J whose elements are "Yand where the norm A of an operator A

DEFINITION from 7^

into

61

4.1.

termed an

sarily equal to its bound will be

ideal

if it

neces-

is not

11

II

A

operators

satisfies the following

two conditions:

A

(i)

^|

implies

YAX

X

Y

for any pair of operators (ii)

VAX

|f

LEMMA STL. T^j

f c

&

Let

4.1.

III

afc,

Y

into Tj!

1

f|A|^=

Proof.

1f-3l

*

g

ates an operator

A

(

and 27:,< Afc)8; Clearly,

A

gc

associate oU

=

tively.

octZ?.,^

)

of

f inife

C(,

g

-norm

^(2^, f tg

$

By Theorem

)

for

on ^Sj/l^with

which

for every (1)

II

A

H^

and

(2).

uniform on

is also

Their adjoints

and

T

S

and

1ft

3.1, this =

4

Ill^llj

=

=

3j

1

gener1

gc

This concludes the proof.

*^^^

Then,

<

"^v

denote two operators on

T

Theorem

l||yl||

expression27L,f t

Let 06 denote a uniform crossnorm on

Proof. Let

and

A from

be a fixed expression. Byfl, p. 55,

-) t

4.2.

T*9^

for which

has the desired properties

LEMMA its

=

.

,

Let^^jf^

^(Z

respectively.

Then, there exists an operator

there exists an additive bounded functional

and

and Tj

denote a given crossnorm on

oC-norm

of finite

on

^gj

||AJI 1/lxlH

III

a fixed expression.

g

(1)

<

||

and

G

are operators on

7i and ^^respec-

T^ and IJi^. For

a fixed

62

expression

oc'(

is

OC.

extreme right

^

G.

!?, F.

By assumption the

IDEALS OF OPERATORS

IV.

in

T??O >^

uniform.

we have,

Recalling that

=

1)1

III

I S

and

III

=

I||T||J

III

of the last inequality is

2, ^<j)

I/I

Sill Illflll

The last holds for every expression 3EJ

f.

g

This by Definition 2.2

.

.

implies ' (

Z,

T(^.

SFj.

)

^

HI

S

III

II/T

III

ot'

(2^,

F.

.

Gj

.

)

This concludes the proof.

THEOREM all

For a crossnorm

411.

Ot

on

1^'^>

operators from 7J> intoT^ (from T^into 1^

ideal

if

and only

if,

first that o(/ is a

an operator froml^ into *^Jof any two operators on 1

2r-

Since O>

is

(

of finite

and

^c\

(YAXf.)8 t

= I

Qirnorm, while

finite

"^

respectively.

<:

IS-^AX^XY*^)!

A

/I

<+

<*>

and

Y

denote

stand for

l|

A

1^

f^

otfZ^x^a Y*

.)

.

g<

|||Y|||

g^

,

=

|l|

Y*/l|

)

is

Definition 3.2 gives

llAl^.

Il/Yll)

implies

X

A

Then,

uniform, the extreme right (remembering

KYAXJl^lllxlU ||

of

-norm forms an

uniform crossnorm. Let

Since our inequality holds for every expression2I^l

Thus,

ot

Banach space

Otis uniform.

We assume

Proof.

)

the

fl

YAX

||

a|

<+

<

,

that is,

YAX

is of finite

Til

IDEALS OF OPERATORS

IV.

-norm and

06

satisfies inequality

To prove from of

T?; into

A

the converse

forms an

)

We remark

first that for a fixed

Lemma

Furthermore, by with

A

||

=

H^

Now,

1

|)

A

of

represents the

IJ^

and

(i)

expressions."^

f^

A

operators

(ii)

norm

of Definition 4.1

g^ all operators

A

assumed by some

A

3.2) satisfy the inequality

4.1 the equality sign is acctually

.

let^JE"^,

8^.

f;

X

b e a fixed expression and

ators on TJ^ and TJ^re spec tively. ator

Banach space

ideal, that is, satisfies conditions

c*/-norm (Definition

of finite

that the

oC-norm (and where

finite

TJ^of

of Definition 4.1.

(ii)

assume

63

By

Y

,

,

any two oper-

remark we can

the previous

find an

oper-

/* for which /I

A

=

/!^

1

Recalling that by

||Y*AX

||

,

Z,(AxyYg t

=

llAXHIxlll

=

^(S^I.Xf,.

Y gi

.

.)

(ii)

w

<

|||Y*|/|

III

Y

Illxlll

III

we have

X Thus, ot

is

11^

(^(ZTT,,^* gj

uniform.

THEOREM

4.2.

ideal of operators

^

III

Y

|X

l|(

III

<*(S..^

g.)

^

determines an

.

This concludes the proof.

For a crossnorm 06

from

^

into

,

(

"^_ (from Tj^into t^

|

)

^U,!?^)

if

and only

if,

OC

is

uniform.

Proof. (

^

*"*

)**

The proof

is

a consequence of

represents the space of

all

Theorem

4.1

and the fact that

operators from

TJ> into

7?^ (from

.

IDEALS OF OPERATORS

IV.

o(,-norm (as was proven in Theorem 3.1).

of finite

"Y*>* )

THEOREM 4.3.

Let

from ^jinto Tjjof

of operators

Banach space

of

preted as the space of

3.1

and

Theorem

an application of

THEOREM

into

,i

T^

^L

4.4.

that is, the of finite

)

Let

and

S

T

,

Remark

3.3,

^

of finite

of all operators in that

oC-norm^approximable Clearly,

it

from

T*>

J^ on

from

p> into

TAS

)

This

fundamental with respect

to

is easy:

<st

.

We

A S

put

Since Ot

is

This implies that also the sequenceSL.-J*. SF/ 9 &

1

mines

TAS

.

.

It

^J

of finite

be an element of

belongs to TJj ^/"^^ since condition

^

o^

7^ (from

denote any two operators on *^ and ^^respectively.

,

4.1 and 4.2).

norm

Then,

T?

norm by operators

into T-i

represented by a sequence of expressions^o

to the

-norm as

OC-norm. Thus,

Definition 4.1 is satisfied even for all operators generated by

is

be inter-

includes all operators of finite rank.

(an operator

sufficient to verify that

(Theorem

may

ideal.

4.2 concludes the proof.

Banach space

A

)***

into "Q^ of finite oc

froml^into T^

Banach space

ideal, then also the

7^ fiL^*

(

the

oC-norm forms an

of finite

Let o^ be a uniform crossnorm

rank forms an ideal.

Proof.

^

operators from

all

well as the space of all operators

*&

oC~norm forms an

finite

operators from T?^into

By Theorem

Proof.

Whenever

denote a crossnorm.

ol/

=

S

F^ <J

if.

**

TG^** d

Gv(**

is

of

T?^**?^)

The operator

.

uniform, OC

(

(ii)

It is

is

in

^O"^>* I

such

(Lemma

4.2).

fundamental relative

can be readily verified that the last sequence deter-

This concludes the proof.

A

CHAPTER V CROSSED UNITARY SPACES Preliminary remarks.

1.

In this chapter

we

investigate cross-spaces generated by unitary and in

To be

particular by Hilbert spaces.

able to

make use

for operators on unitary spaces, the following

Let

of

some known theorems

comments are

vital:

be a complete unitary space, hence a Banach space, and as before

fc,

Fv stand for the space represents the norm.

of additive

Let

f

bounded functionals on K/ where the bound

FC/

s

be fixed.

Define

by means of the

f

relation: =

f*(
is

f

f

(if.

for

)

an additive bounded functional on pC

,

Pv

.

.

with a bound equal to

II

f

ll

Conversely, every additive bounded functional on ^/ can be represented in the

above fashion uniquely. 9\s

and

*

The one-to-one correspondence

f

f

^""*

between

fvobviously satisfies the following relations:

* fj

^-

i

^r->

f

X ^^

and l

*

-*

implies

^

fj

+

f

^

^pf

+

f^

.

However, ^.

f

For

this

implies

af

j^

j^

reason the correspondence Wj

isomorphism.

p(/

is a

^ "af

H f

+z

for any f

is

Banach space. Moreover

complex number

a

referred to as a conjugate f

^

^*

f

implies

66

II

CROSSED UNITARY SPACES

V.

f*|| =

f

||

Defining however,

.

)I

=

(**>f) we introduce an it.

inner product in Pv

we

In the future

shall denote by

g

(

>

and

,

*

)

II

f

is the

II

f

Thus, with Pv also

f^

morphism

f\/ is

f

,

elements will accord-

its

,

a unitary space.

Despite the conjugate iso-

between them we shall consider them conceptually

f

5p2T

<

,

"g

,

that goes with

the space Fw to which an inner prod-

Pv

uct has been added in the described above manner; ingly be denoted by

norm

distinct.

Now

let Tp,

As customary we form

gate spaces.

F 3 G

and'^^denote two Banach spaces and

F G^c^, G eTr- B /

for

C(?

ip.^

,

fv,

into Jp

^

'"f

(<^>rp)f

=

We

.

F\/

,

(^>

f

additive operator on

from Pv

)f F>/

p\/

,

.

of

rt,

p>/

.

may be

'f

It

.

9

*X^>,

their conju-

T^^X

number

and

,

F(f)G(g)

has been also sugas the operator

g

Since

,

With

Pv

may

.

Due

to the conjugate

isomor-

be also interpreted as a conjugate

desirable to deal with additive and not

it is

FC"

G(g)f

interpreted as the operator

into

^

the
conjugate additive operators on Pv

^!Q, g

Definition 1.3, the

f

,

follow this pattern in the case of unitary spaces.

the <>

phism between Fv and

of

f

-- although not essential -- to interpret it

from

for

g

(F0G)(f0g)

represents their inner product gestive

f

7^

we are

led to consider

Pv

Pv

instead

Accordingly we shall form the inner product for elements

Pv by elements

^

(9

*f of

pO

Fv

conforming

conventions: (

f <0

g)(
f)

=

(f

,

f )(

g

,^)

.

to our

previous

f

g

V.

Both

f

<8i

is clearly

9 y may

and
g

CROSSED UNITARY SPACES

an operator on Pv

hence also an operator on Fv

be interpreted as operators on

is

and

Thus, both f^/0 fi

Pv

first

where

,

=

(
^

an operator from

(Cf^t)f

The

pj/

defined by

,

(F*g)
67

for

g)f

= P>/

into

=

Cf^fv.

P{/

for

(f.rj/)
Pv are isomorphic

,

defined by the relation Cv

f

.

to the linear

space of oper-

ators of finite rank.

The canonical resolution for operators.

2.

Throughout the present Chapter we shall assume that

p\/

represents a

Hilbert space (unless otherwise specified). All operators considered here are (unless otherwise explicitly stated)

and their range contained in Pv

composition given by

L,emma

'J.

.

assumed

to be defined

To these we

von Neumann in 22 J

,

shall apply quite often the de-

and expressed below in

5.1.

For our further discussion we operator

*A

is

Hermitean and

If

=

A

d*A

.

We

is of finite

definite.

LEMMA

write symbolically rank, then both

5.1.

isometric operator

Let

W

A

Following 22] or [21, p.

It

AA

B

=

abs(A)

and

initial set is the

B

Fv/

,

or

B

.

the 249]]

such that =

(/?*A)

are also of

abs(A)

denote an operator on

whose

A

recall that for every operator

there exists a unique Hermitean and definite operator

B*

on the whole space fv

finite rank.

There exists a partial

closed linear manifold deter-

68

CROSSED UNITARY SPACES

V.

mined by

the range of

such that, the following formulae hold:

abs(A)

W

(i)

A

(ii)

abs(A)

(iii)

abs(A*)

(iv)

abs(A)

=

abs(A)

= = =

.

W*A

W

.

abs(A)

W*W

abs(A)

The decomposition presented

A

W

=

B

where

W*. .

in these

formulae

and

W,

with the initial set the closure of the range of

B.

sense:

and

W

=

W|

B^O

f

(

.In

the case

A

isometric

is partially

,

implies

rank we

is of finite

unique in the following

is

=

B|

may assume

abs(A)

W

that

is unitary.

Proof. First we prove the existence of the above decomposition. .

jg

AA

is

and

B*B

Her mite an =

B*~

definite on =

A*A

=

Putting

C

C

Ajt

and

Bibs(jf)

,

=

.

B

For

=

abs(A)

tf.

B

we have,

=

=

||

||

Bf

we

(Ajf)

||

.

obtain a similar relationship between

namely, =

llA*f|l

flCfll.

possible to construct the partial isometric operator

W

whose

Thus,

it is

initial

and final set are the closed linear manifolds determined by the ranges

of

B

and

A

B

Hence always

.

Af

II

pv

,

respectively, and such that

A

=

WB

C

=

WBW*

=

CW ,

,

A*

B

This proves the existence of the stated decomposition.

=

=

BW* W*bW

W*C

=

.

CROSSED UNITARY SPACES

V.

To prove

W

and

Bj ^-0

A

the uniqueness, suppose

as well as

WB

=

69

=

W. B.

are partial isometric with the

'Vj

The

respectively.

determined by the range

A*A

B

BW*WB

= =

=

A

ranges of

and

.

BWWB

finally that

Suppose

A

of

and therefore

B,

W

final set of both

B

and

W.

A*

Then,

=

W

=

A

is of finite

f

closed linear manifold

BW*

=

B

B~

B*" =

=

>f"A

B

and

is the

J

that is,

,

initial set

B

being the closed linear manifolds determined by the ranges of

B ^

where

f

,

V^*

hence

,

W, rank.

Since

are of the same dimension.

w'

dimensionality the isometric transformation

Af

||

Due

=

JJ

||

Bf

the

11

to this finite equi-

on

Bf

of all

all

one-to-one, and possesses a unitary extension (not unique however!).

Af

is

This

concludes the proof.

A

The unique decomposition

W

=

abs(A)

proven above for operators

will be referred to in the future as the "canonical resolution*.

3.

A

characterization of the completely continuous operators.

LEMMA lp0Bo|/

5.2.

^

Let

,

rjt

as an operator on fv

,

denote two elements in fv

,

whose defining equation

=

(Cf>^/)f

for

(f,rf/)
.

We

consider

is

f

fC

Then, (i)

(

(ii)

a

^
=

=

a(

~ ("')

~

a '

denotes any complex number)

.

70

CROSSED UNITARY SPACES

V.

1

(iii

)

(iv)

A(

(v)

denotes any operator). (v')

Proof. of

/<>

These^relatio|iships are a simple consequence of the definition

and

p

be verified immediately by an elementary calculation.

may

Similarly, the meaning of the symbol

LEMMA

Given two nos

5.3.

negative numbers

and

vP-)

which

for

a^)

(

(

!i^ a -^#

(f^>)

r>f;

s

clear.

and a sequence of non-

=

lim a^

^

^

Then, the infinite

series

is

convergent for every

in

g

pv

A

sum determines an operator

,

A Its

we denote which we

=

sum

its

Ag

by

.

Proof. For

Thus,

We

Ag

n

is

evaluate

S^C^-tc

||Ag||

=

A =

a

sup

-

c

we have,

defined for every )||

V

>m

A HI

.

|//

S,

the other hand putting <^p^ for

For

%

I

g

g

(|

g

[l8, = j|

(*,r we

get

Thus, the

shall denote symbolically by

bound 111

On

;

Theorem 1

,

4

1

,6J

we have,

.

CROSSED UNITARY SPACES

V.

=

=

||

llA/fJ)

at

a^CpJI

for

=

i

71

.......

1, Z,

The last two relationships prove that ill

A

=

HI

sup

||

=

AgJJ

sup *

"gil*i

a,. u

.

This concludes the proof.

LEMMA is

abs(A)

An

5.4.

A

operator

is

completely continuous

and only

if

if,

completely continuous.

The proof

Proof.

is a

consequence

In our future discussion

we

of

Lemma

and

5.1 (i)

(ii).

shall avail ourselves of the following charac

terization of the precise class of completely continuous operators:

LEMMA

exactly those which have the "canonical representation"

where

(

C.

)

and

(

are orthonormal sets and

f^)

lim a^

Furthermore,

positive.

=

has an infinite number of terms.

form

A

The completely continuous op-rators

5.5.

in

on f^ are

all the

case the sunn

!3

The above representation

^ is

Let

A

be completely continuous.

also completely continuous.

Hilbert)

it

Since

it is

f^f

of

are

^4% &+&unique.

proper vectors of

Lemma

5.4,

abs(A)

there exists a complete

is,

abs(A)

.

Let

a

^

.

a^

abs(A)

a

, (

a^

,

.........

denote the corresponding point proper values (with multiplicities), that

The definiteness implies

The

also Hermitean and definite, (by

possesses a pure point spectrum, that

orthonormal set

By

rK-

j,(?

a^*s

the positive point proper values (with multiplicities) of

Proof.

a

.

*g^

The complete continuity implies

is,

is

lim

CROSSED UNITARY SPACES

V.

72

=

positive

where

a

s

^V a*1fe* tk l

we

Relabeling the subscripts

-

(.'

are positive, and the

a Js

all the

=

abs(A)

Clearly,

.

a^

By Lemma

5.1,

A

=

Wabs(A)

closed linear manifold determined by

form a nos.

W

where

,

(np?

.

)

consider the =

abs(A)

get

(ffa)

We

*

isometric on the

is

Hence,

(W<^/.

forms a nos.

)

Thus,

A

=

w(

numbers we have r4/.

the

,

)

equal to

sum

Z

t

=

lim a^ 2EL { a

sup a^

.

.

^) & pi/. t

=

at

wrftrpc a

Suppose that for a sequence

Conversely.

(

=

^. t vy.^)

By Lemma

,

5.3, for

a

2l t ,

represents an operator

( ^pj.

A

fi

of positive

....

two nos

cf

t
and

)

with a bound

Since,

A -

III

A

the operator finite rank.

Since, that is, the

can be approximated in bound by a sequence of operators of

Thus,

^A a.*s

A =

must be completely continuous by :JT c

a^ef

,

we have

abs(A)

1

,

=

p. 96j

.

StV^fli^.

represent the positive point proper values of

abs(A)

This concludes the proof. Quite often completely continuous operators (and therefore also those finite rank) will be

4.

The Schmidt-class

LEMMA (o|/.)

,

represented in the "canonical form** given by

5.6.

the infinite

5.5.

of operators.

Give$ an operator

sums

Lemma

of

A

on

f\,

,

and two cnos

(

(p.)

and

CROSSED UNITARY SPACES

V.

of

73

non-negative terms, are either absolutely convergent or properly divergent,

have well-defined values. For these sums the following

at any rate, they ail

They are equal

statement holds:

*

Proof. |*

=

Zc,

Clearly,

2-d

=

each other and independent

=

(Arfy,<.)

l(A(fl,,r)r

A fjr

i

to

(A

,

/-f

.

.

)

of

By Parseval's

(

<#),

(

^

)

.

equality

Hence,

-

z cj^ ^ ,r

=

.

z^

(A*^,

i

fi)

r = z, H AV

This concludes the proof.

DEFINITION three

sums

L,emma

in

We

5.1.

5.6.

the E. Schmidt-class

LEMMA

denote by

<^(A)

(

A

The operators

(sc)

the

)

for

common

which

6"*(A)

value of the

< 4"

.

5.7.

(i)

A

(ii)

A

(iii)

A

(iv)

A

= (sc)

Gs>

(sc)

B

,

^

&

(sc)

A*

S

> aA (sc)

&

->

^ AX

(sc)

(

.

sc )

(A

-f

(

B)

tt.

XA

and

a

i

(sc)

&

a ny

s

complex nunnber).

.

(X

(sc)

denotes

any operator).

(()^p

(v)

(iii):

and

<^(AX)

(iv')

Proof,

form

e.

(i), (i*), (ii)

((A

+

d(XA) for

(sc)

^ (p

J||

,

X

rf

|/|

in

tf(A)

Pv

.

.

are obvious. -

(A
B)ft,^.)

^

2(

-f

(B^.^.)

hence

CROSSED UNITARY SPACES

V.

7^

sum

Thus, the third

(remembering Now,

)/

5.6 yields the desired result.

XA

suffices to consider

It

(iv), (iv'):

Lemma

of

and

(i), (i'),

|||

XA
X

=

l|l

1I|X*U|

=

.

)

Therefore the first sum

.

A^pJ

fl

AX

since

,

of

Lemma

5.6

furnishes the desired result. (v):

For a cnos

By Bessel's It is

(r

inequality the

we

)

have,

extreme right

a consequence of (v) and

is

^.

above that

(iii)

||

||epH"*'

all

VP

(I*"

operators of finite rank

are in the Schmidt-class.

LEMMA the

sum

2*c

(Aip.

The result follows by considering Independence of

(we use is

7R,

Lemma

unit)

(

(ii), (iii),

VP.

we see

5.2.

5.8 is denoted by

and a cnos

(sc)

~

B(#)|^

sum

the first

=

)

^C

(

*(A

A

*) 21^

(A

,

of

Lemma

+ B) J*"-

and Definition 5.1).

that

For

(A
|

Replacing

)

in

(

If

AWjl*

(

<&)

(<*pc

+

)

A ,

B)

(

B

,

.

by

iA

A
B ft

in

(sc)

i

(

)

is

.

5.6 and Definition 5.1.

(tf(A-B))*)

Thus,

,

llB^J)

:

'

5.7

DEFINITION

Lemma

)

ST^f, B fc

independent of

imaginary

(D.

(

B

,

absolutely convergent and independent of

is

Bcp.)

,

Proof. Absolute convergence:

Therefore,

A

Given two operators

5.8.

*2R.2L L (A
,

,

B<

stands for the

independent of

the value of the

sum

( *ft. )

of

)

CROSSED UNITARY SPACES

V.

LEMMA

75

5.9.

(i)

(B

(ii)

(aA

,

=

A)

=

B)

,

(A

B)

,

a(A

.

*

B)

,

L =

(if)

(A

(iii)

(A

(iii')

(A.B.+ BJ

(iv)

(A

,

A)

2>

(iv')

(A

A)

=

,

,

aB) +

a(A =

A^B)

t

+

(Aj.B)

=

any complex number).

is

J

B)

,

a

(

(A,B

(A% +

)

,

B)

.

(A,BJ

(

.

.

A

only for

=

(v)

(XA

(vi)

(AX

(vi')

Proof,

(A

,

B)

=

(A

,

for all cnos

A

(

(ft)

=

,

(iv) this

A

or by

-

(vi):

B

-I-

B)

c) (A

(A<jf.

.

,

sum

A
,

= )

1/AcpJI

Lemma

of

A
becomes

Lemma

7R,(A

denotes any operator).

=

.

^(A)

5.6,

for all

A (A.

means

=

llcp//=

1.

=

(

This

.

By

A

X

(

f

BX*)

that is,

equation holds by

by

^I

are obviously a consequence of Definition 5.2.

Using the first

):

(v):

=

(i)--(iii')

f

means

,

B)

Obvious, since

(iv):

(iv

,

B)

-

B

Hence,

Obvious, since

(J?

,

=

,

we obtain

iA

by

^)

(XA^pc

A

=

and subtracting, we obtain

Next, replacing n*A .

for

B

.

The last

A

Replacing in this particular case

5.7 (i*).

A

g (A)

=

(j(A*)

(A

BcP-

,

B) =

)

.

$

ffl,(jP

(/{*,

B*)

,

B>) =

B

=

=

,

76

CROSSED UNITARY SPACES

V.

(vi'):

Replacing in

(v) to

applying

5.1.

Lemma

(ii)

X

,

and

we

(iii)

5.9 (i)--(iv')

J?

by

B*

,

X**

,

and

,

obtain the desired result.

state that the operators in

makes

it

clear that

(A

,

B)

(sc) is

and

(sc)

(A

norm

5.7

Lemma

linear space.

inner product in

the

B

.

both sides of the equality,

REMARK form a

A

(vi)

that goes with

it.

=

A)*"

,

6T(A)

Therefore, we also have Schwarz's inequality:

[(A ,B)|

REMARK operators of

B

and

=

5.2.

finite

Z.

:

<7

coincides with Definition 1.3.

By Parseval's

rank hence

m f,9 .

O

In the particular

g.

,

in the

A

case when both

Schmidt-class, say

then the inner product

(A

,

and

A

=

B

are

2lTl.
defined above,

B)

<&

(ZT,?j To see

g,)(

2I7C W&rf.)

this let

(

identity [18, p. 10]

c*X

)

"

the one stated --

denote a cnos in ^v

in

Then,

:

Thus,

This concludes the proof. In conclusion of this section let us add that

Schmidt-class

and

[i6j

of

some properties

operators have been investigated in

2CfJ

,

of the

and later in

l5]

an

CROSSED UNITARY SPACES

V.

The trace-class

5.

LEMMA B

C

and

of

operators.

are in

(sc)

and a cnos

,

absolutely convergent, independent of

Proof.

REMARK in

(sc)

B

B

C

,

Lemma in

,

,

C

by

5.10

(sc)

C

,

may

of

are in ,

5.10 by

By

[22, p. 302]

By

the

,

(tc)

t(A)

nite, for a given it is

cnos

For such an

.

=

A

=

.

Then

<

Lemma

A we

(A A)^

ArA

is a

this is also true for

(

of

to

C)

,

is

)

.

B

(for

C

,

5.7

BC

,

C

,

separately.

A

form (i),

=

C**B

and replacing

A

considered in

with

B

C

,

,

)

,

the

sum

is

Remark

5.4)

denote the value of the

sum

5.10 (or of

A

and term the "trace" of

,

abs(A)

same argument

A

The operators

5.3.

Consider an operator

hence

C)

Lemma

A

,

.

the trace-class

Lemma

,

,

evident that the operators

it is

,

Considering

.

where

5.8 and Definition 5.2.

(B

B

(A^>.

(B

5.10 deals with operators of the

(sc)

B*

sum2.^

Lemma

only, and not on

Cr B

=

and equal to

(<&)

be characterized in the form

DEFINITION form

Lemma

5.4.

the

,

(fj. )

5.10 also shows that

(TB

depends on

)

(

A

such that

simple consequence of

Lemma

5.3.

REMARK where

is a

This

A

Given an operator

5.10.

77

a

Hermitean

definite operator.

uniquely defined definite operator. (abs(A))*"

.

,OpJ.

Since )

is defi-

abs(A)

has ~p

terms,

either absolutely convergent or properly divergent, at any rate

a well-defined value.

it

has

CROSSED UNITARY SPACES

V.

78

The cnos

in

v#.)

(

Lemma

5.11 (iv) below, should be

trarily given but fixed, i.e., the criterion is valid only, and

it

does not matter how this

LEMMA

5.11.

(i)

A

(ii)

abs(A)

(iii)

(abs(A))*

(iv)

Z

Proof.

(iii)-*(i):

*

W(abs(A))*

e

=

proves

abs(A)

(iv)

=

(I

1"

^s (sc)

LEMMA

.

5.12.

fe

(sc)

W(abs(A))

,

By Lemma

.

^

.where

(abs(A))

B

,

W*B S(sc)

and

*

=

C

=

Wabs(A)

are in

by

,

5.7 (iv),

(sc)

Lemma

.

AC

(tc)

Then,

5.7 (iv).

This

.

This :

that

BC

=

W*fcC

(tc)

each other:

to

.

Hence,

A

/

(abs(A))

(sc)

(abs(A))

r

chosen.

.

Let

Let

(ii)-* (iv):

(tc)

show

W*A

abs(A)

e

.

(

.

shall

(sc)

(i)-^(ii):

is

tp.)

applied with one cnos

if

The following statements are equivalent (tc)

We

(

viewed as arbi-

immediate by Definition

is

=

.

(abs(A))
fc

<

(abs(A) <.,

I|

sum

Thus, the first

-f-oo.

and

5.3

Lemma ,

5.10.

hence

Cf^)

of

Lemma

5.6 yields

This concludes the proof.

Let

(i)

A

<=:

(ii)

A

6. (tc)

(iii)

A

,

(iv)

A

^

(tc)

B (tc)

a

denote a complex number and

A* e.

^

-* aA (tc)

^

.

(tc)

(tc)

-* A

AX

^

+

B

(tc)

.

.

E

(tc)

and

.

XA

<.

(tc)

.

X

any operator.

CROSSED UNITARY SPACES

V.

Proof, to

Lemma

Obvious by

(i):

A

since

(i),

BC

=

is

equivalent

A*

AX

Lemma

(ii):

Clear, by

(iv):

Immediate, by

=

By Lemma

Lemma

5.11 (iv),

it

5.7 (ii).

Lemma

XA

and

B(CX)

(iii):

=

(XB)C

+

A

(tc)

B

,

-I-

are in

Thus, by

.

(W*B

(tc)

<.,<))

T.

= (p.)

.

Therefore by

.

+B)C,

(iv)

W*A

above,

(W*A.,

S/(abs(A

B)
-I-

A

In (i)--(iii) both

B

,

are in

(tc)

either

A

Lemma

5.12 (i)--(iii); the expressions in (iv) are defined by

or

B

is in

=

(i)

t(A)

(ii)

t(aA)

(iii)

t(A

(iv)

t(AB)

Proof,

(i)--(iii):

(iv):

Following

-I-

Assume

Lemma

5^

,

(f>.)

<

(\V*B
,

(p.)

also

This concludes the proof.

absolutely convergent.

5.13.

\V*(A + B)

Zlc (W*(A

are absolutely convergent and therefore

LEMMA

=

B)

This implies that the sums

.

implies

suffices to establish the absolute convergence of =

Now,

BC

=

.

abs(A

5.1,

A

5.7 (iv), since

(ft)

in

5.7

79

(tc)

t(A*) =

-

+

t(A)

t(BA)

put

Lemma

5.12 (iv).

t(B)

.

.

Obvious, by

we

are defined by

denotes any complex number).

a

(

Lemma

by symmetry, that

5.10

in (i)--(iii)

in (iv)

.

at(A) =

B)

The expressions

.

;

A

=

5.10 and Definition 5.3.

A^ C?*B

(tc)

,

X

and write

where

B

,

C

for

are in

B (sc)

.

By

CROSSED UNITARY SPACES

V.

80

Definition 5.3, =

t(AX)

=

t(C*BX)

t(XA)

=

t(X C*B)

(BX =

C)

,

(B

,

CX*)

.

The two extreme right-hand expressions are equal

Lemma

5.9 (vi')

.

DEFINITION the last

number

LEMMA

each other by

to

is

A e

For

5.4.

defined by

5.14.

In

(tc)

Lemma

5.11

define

(ii)

A

what follows

we

,

t(abs(A));

.

B

,

=

m(A)

are in

(tc)

,

and

X

denotes

any operator.

Proof,

The extreme

Lemma

Thus, by

Obvious, since

may

Lemma

5.1 (iv)

and

|llX*U|

It

=

abs(XA)

suffices to consider

I/I

X =

|J|

)

.

Lemma

By

W*XA

XA

,

be written as

m(^)

^

|a]

=

,

^A

=

abs(A)

since

AX

=

t(abs(A)\v^W)

5.1 (iii).

by

Therefore,

.

/a/

Lemma

m(A)

.

=

5.1 (i),

A

by

t(Wabs(A)W*J

X

=

(aA^(aA) abs(aA)

(v):

=

t(a.bs(*))

right of the last equality

5.13 (iv).

(ii):

=

m(A*)

(i):

Wabs(A)

(X*A*)*, (we use

(i)

CROSSED UNITARY SPACES

V.

where

Y

=

=

J|)w|||

W*XW (

sc )

=1

fijw,*!)

Therefore,

.

(abs(A))*"

and

I

III

Y

Hence,

.

^

IJ|

l||x|J|

Lemma

Consequently,

81

=

abs(XA)

Now, by

.

Lemma

5.1 (iv) gives

with

Yabs(A) 5.1

1

(iii),

Y(abs(A))*

&

Therefore, =

m(XA) t(

=

t(abs(XA))

=

Y(abs(A))* (abs(A))**

((abs(A))

t

)

f)

(Y(abs(A))*-

,

Using Schwarz's inequality (Remark

^

m(XA) Thus, by

5.7

((abs(A))

(i*)

and (iv

|||x)||t(abs(A))

-

4 ,

=

Recalling that

((abs(A))*-

,

*

1

-

=

.

t(W(abs(A))*.

extreme right we

) (3((W(abs(A))'

and using

4

m(A)

f)

to the

(abs(A))*)

t((abs(A))i (ab^A)) t(abs(A))

m(A)

(W(abs(A))*

t vf*l}l

=

5.1 (i),

S'((abs(A))' |||

=

)

)

Illxlll

Applying Schwarz's inequality

$

get

)

t

t(Wabs(A))

((absfA))

|t(A)|

we

f

(abs(A))

,

By Lemma

t(A)

5.1)

4

4

|x(t((ab8(A))*(ab8(A))

(vi):

.

rf((abs(A))*

Lemma

ITIxM

=

t(Yabs(A))

=

) .

t

f)

Lemma

5.7

get,

.

(i')

and

(iv*)

we have,

(sc)

82

CROSSED UNITARY SPACES

V.

Formulae

(iii):

A where abs(A

=

Wabs(A) =

Blwffl

m(A

1

=

+ B)

lllxlll^l

(i)

=

W,Jl|-*l

,

Lemma

of

Y |B*1 =

+ B)

.

+

Lemma, we

+ B)

X

=

t(Xabs(A)

+

W%W

=

Y

,

W^

;

Now, =

t(Yabs(B))

Yabs(B))

=

.

Applying Schwaris inequality and this

W^(A

Therefore,

.

where

=

abs(A + B)

,

1

wjfl

111

t(abs(A + B))

t(Xabs(A))

5.1 furnish

Wj abs(B)

Xabs(A) + Yabs(B)

HI

.

(ii)

B

,

HI

,

and

(i*), (iv)

of

Lemma

5.7 or (vi), (v) of

see that the extreme right of the last equality

is

not greater

than

(iv):

4

m(A)

fllxljl

m(B)

|||YH[

=

t(

=

a bs(A))

t((abs(A))*

1

((abs(A))*

5.9 (iv

^

%

)

)t

AA

This implies

LEMMA

it

,

+

m(B)

.

=

,

m(A)

( tf((abs(A))* =

(abs(A))"^

and therefore

'=

(abs(A))t

)

)

)*

implies =

,

A

that is,

.'

g((abs(A))^-

= )

(abs(A))*

=

5.15.

(ii)

H

(iii)

m(

Proof.

and

follows

ff

(i)

=

(abs(A))*)

,

m(A)

Therefore,

trivial.

m(A)

Clearly,

m(A)

By Lemma

<

fe

(tc)

We may assume

By Lemma

5.1 (ii)

that

and

(v),

and *^

*>

otherwise the proof

is

CROSSED UNITARY SPACES

V.

Now, put

*P

m

Then

J0

I)

=
1

and

,

^

H*! H

operator corresponding to the proper value

which

orthogonal to

is

Then,

(fj

,

,

that is, to

,

to the

CP

,

Cp

proper values

cnos

f.

^

)

correTherefore,

.

corresponding

abs(CP$rp)

Consequently,

.

=

(

Cp&r

,....

,....

,

to a

(abs(

,

are also proper vectors of

,....

^

extend

II'Y/I,

lltfll

Further, every

.

proper victor of the last operator

are proper vectors of

^, ^"--

sponding to the proper values C0

fl^pl

B^plJ

We

.

a proper vector of the last

is

is a

,

*^f

proper value

to the

corresponding

tp|

83

i

abs( for

We

have,

Thus, by

m(

i

Lemma =

>^p

II

)

To prove

This proves

abs( Vp

(ii),

we

r^

)

e

This proves

take any cnos

and

(tc)

(i)

and

r^

G.

(tc)

.

Moreover,

(iii).

Using Parseval's equation,

(^N^)

This concludes the proof.

(ii).

REMARK operators of

5.6,

5.5.

(tc)

It is

form

a consequence of a linear set.

tains all operators of finite rank.

Lemma

Because

Lemma

5.11

of

5.12

(ii)

Lemma

proves that

and

5.15

A

(iii),

(i),

in

that the (tc)

con-

(tc)

is

8^

CROSSED UNITARY SPACES

V.

equivalent to

abs(A)

(tc)

prove that

(ii), (iii), (iv),

(Lemma

in

therefore

;

is a

rn(A)

norm on

Lemma

is defined.

m(A) (tc)

crossnorm

in fact a

,

5.14

5.15),

6.

Symmetric gauge

functions.

In this section* we present

functions which

we

shall

some elementary

make use

results for

These

of in the future.

symmetric gauge

will

permit us

derive later an explicit representation for all uniform crossnorms on as well as

some

,

to

0v

First we investigate gauge func-

relationships between them.

tions on a finite dimensional space.

DEFINITION of

space tion

if it

A

5.5.

n-tuples

(

real function Uj,....,

^>

(

of real

u^)

u ^...jU^J

on the n-dimensional

f

numbers

is

termed

a gauge func-

satisfies the following conditions:

(u

(i)

(ii)

<J

(iii)

$(

<1)

(

will be

,....,

>0

u^)

cuj,....,

cu^)

u,+ u;,....,

=

unless |

c

(

>

....

u^,)

$(V- ^)

Hj-iO^

termed symmetric

u

y(

|

u,

if i

n addition

=

u^=

.

for any constant +

c

.

$(<,...., u^)

to (i), (ii), (iii), it satisfies

the following condition: (iv)

(

Et s il

where

u t

,....,

and

1

=

U.J ,....,

(^(^^,....,^1^ denotes any permutation on

n'

To simplify our formulae we

shall always

the following condition: (v)

$(1,0

......

= )

1

.

assume

that o>

1

,

,

n

.

also satisfies

CROSSED UNITARY SPACES

V.

A symmetric |uj4l u cj

that is,

$(;

u',

LEMMA on Vv

It is

p^

^b

5.16.

Then, for

Proof.

By

for

This

.

u^)

,

gauge function

is

Let ^1

=

i

a)

p.

(

$

virtue of (iv)

(

^

for

u

.....

>

u

,

p

<.

1

1,....,

u

,....,

The

implies

^

(

each variable u

,....,

u^)

u/

,

^

Lemma:

denote a symmetric gauge function

u^J

we may suppose

that all the

u.*s

are

^

sufficient to establish the last relation

i

U u ...... U *' P ; tt,

.

n

in

we have,

1

it is

occurs only for one

$

non-decreasing

precisely the content of the following

clear by induction that 1

is

85

,

.

when

that is,

J

3

$

(

last assertion follows

uiH u

u i-

from

f U UH' -

HJ

the following simple

direct calculation:

u,

REMARK that

Cp

5.6.

In the proof of our

...... u- ......

Lemma we

satisfies condition (v) of Definition 5.5.

uj

have not assumed at

all

86

CROSSED UNITARY SPACES

V.

LEMMA function on

ft( u

Let

5.17.

{

Lemma

Proof.

......

f>(

and

(v)

equals I

$.

,

^$(

u

^(

|f

.....

.

u^)

5.16 gives,

,

......

u^,

)

4.

of Definition 5.5 for

(ii)

(u^|

UJ

denote a symmetric gauge

u^)

K^. Then,

mjxfuj

By

,....,

f>(

V

uc

u^,

,

u^,....,

uj

.

the left side of the last inequality

that is,

u

..... ,

V'" uiJ

for

i

=

1

n

,....,

.

This concludes the proof.

LEMMA we

5.18.

For a symmetric gauge function

ft

(

u^

,....,

u^

on

have,

The proof

Proof.

and

(v) for

LEMMA

5.19.

A

last is

We

A

(

u

|f

gauge function

^(

and

m

Conditions

If <,-; ...... The

a simple consequence of conditions

(iii), (iv)

.

<3^

Proof.

is

^ 2L^J u

" i

(ii)

(iii)

for

u|f ...., u^)

is continuous.

furnish

vol u cJ

^ Lemma 5.18.

This concludes the proof.

are about to define an associate of a given symmetric gauge function

....,

u^)

on

k^

.

For a fixed

n-tupie

(v

f

,....,

y^)

,

CROSSED UNITARY SPACES

V.

8?

1

(a)

a? represents a continuous function on the compact (that set of

n-tuples

assumes a maximum which we of

course

may

set of all

The proof

) (

u

,

____

,

u |t ...., u^)

5.20.

^X7(

is such.

Vj ,....,

If

+

is a

v^) is

(f)

Lemmas

-f

(u^j

LEMMA an

n-tuple of (1)

Furthermore, (2)

5.22.

n-tuples

(

ir

(

u

,

____

u^J

,

(

F(u |f .....

Vj

u

,

u

,

,

=

u

|V|

+

....

bound

V|

F

on

jj

v^

,....,

on which there

)

^%

^(

P(

(

(^

&))

.

determines

)

+

\J

for

(

u,

......

Ul|

)

in

K^

***

max

-

|F

J

for

such that,

,...., v^^)

uj

last

over the

(a)

will be denoted by

Every additive functional

numbers

its

(h

The

.

symmetric, the same holds

.....

defined a gauge function

y^)

it

gauge function whenever

i

is

Hence

.

immediate:

is

U|

of

1

.

)

5.21.

The n-dimensional space

=

numbers

of the

,....,

(

....

TJ/( Vj,.,..,

maximum

^

two

of the following

u^

LEMMA

(

|uj

shall denote by

be also defined as the

n-tuples

LEMMA

which

for

(u,,....,u^)

closed and bounded)

is,

u ,

where

the last

max

is

extended over

Conversely. Given an

n-tuple of

all

(u,...., u

n-tuples

numbers

ship (1) above, determines an additive functional

(

F

v

,

.....

(

.

Its

v^)

bound

)^fe(0,....,

,

0)

then relationis

given by

(2).

.

CROSSED UNITARY SPACES

V.

88

F

Let

Proof.

=

F(l,0, ..... 0)

be an additive functional on

F

of additivity for

=

F(0,0,....,l)

v,,....,

.

v^

Then,

.

(p (1)

(

A

Put

.

)

above

is a

consequence

Clearly, (2) follows from (1) and the definition of a

bound for an additive functional. That the converse holds,

immediate. This

is

concludes the proof.

LEMMA

This

Proof. a given

<3p

5.23.

is a

5.6.

<J>( u,....,

LEMMA same time

5.24.

of

K^(^$

Lemma

be characterized

may

)

5.22 and the definition of

"VfT

(

v, ,....,

termed

is

v^)

"Y

for

the associate gauge func-

.

u^)

Any gauge function

2>

(

u

,

,

;

Lemma

(p ("^)

5.23, the conjugate space of

But every

Therefore, the conjugate

of

finite

K^C^O

on

u^)

the associate of its associate gauge function

Proof. By terized as

consequence

of

.

DEFINITION tion of

The conjugate space

*"\j/

Q

(

(

rf)

y*J^

v

,

____

,

f

)

be characterized as

v^

is

p~

.

be charac-

may

dimensional Banach space

may

is at the

reflexive.

(

A)

This concludes the proof.

Consider the set whose elements are (

U|, u., ....

tion of

fashion

)

sequences

of real

numbers

having only a finite number of non-zero terms. Defining addi-

elements and multiplication

we

infinite

obtain a linear set \^

.

of an

element by a scalar

in the obvious

CROSSED UNITARY SPACES

V.

DEFINITION

5.5'.

Under

shall understand any function

conditions

elements

(D

yjj

on (^ we

symmetric gauge function (^ (

u

,

u

,

defined on y^ subject to

)

(

-- (v) of Definition 5.5 in which the

(i)

of

a

8

n-tuples are replaced by

that is, infinite sequences having only a finite

,

number

of

non-zero terms. Clearly, the set of all elements u

=

m

u

(

$(u (T)

(

u

,

1

,

,

LEMMA

^( Proof.

vj

v,

common

Their

we

value

By

)

=

TJTJ

v

(v

*" )

'

Thus,

u

,....,

u

,

u

,

,....

,

A

Vv

=

n

,

v,

(

,

=

v^,

)

^17

(

v

,

f

C)

with

(

u^

,

,

u^)

=

Each v

v/w)

|

we have,

)

^ ,

@

given gauge function

"^Kj

v

of

)

1,2,

v

x(

v^,

V

v_,

,

,

,

,

)

)

we have,

max

=

u v -

+

-

^r _

l_+Jja

'

To prove the converse inequality, we remark by values

,

K/^ an associate

shall denote by

definition,

u

....,

on fc^ for

For a given n-tuple

5.25.

|f

defines a gauge function

determines on

u^)

,

;

,....

,

u

be identified with

on y^

)

u^,

,

may

,

,....

f

u

n

L^

for

(

we have,

Lemma

5.16, that for any

n

-I-

1

CROSSED UNITARY SPACES

V.

90

u v

^

Definition 5.6 implies

The rest

of the

( (

v, ,....,

is clear.

proof

-I-

&J

V

)

v^,

....

t

t

^

-H^v^,

.....

TfTj

nJ

O

v ,

Thus

'

This concludes the proof.

V

LEMMA on

)*J

the

,

*\j7

Proof.

For

5.26.

(

v

,

v

Clearly,

in the

u

(

,

N

,....

symmetric gauge function on

(i), (ii), (iv)

and

of

)

(*

Thus

(v).

Given two sequences representing elements

(iii).

u

it is

(v

such that for both sequences the terms with the sub-

are equal to zero. This means, the sequences

may

be written

form v f

We

is also a

)

satisfies properties

\JT

there exists a number

N

.....

symmetric gauge function

f

sufficient to verify

scripts C->

a given

,....,

v^

,

and

.....

,

v^,....,

v^

,....

,

,

have,

v,,....,

VH

,

o

.

o .....

)

y

4-

'

(

V|

^

......

.

o

,

o

.....

This concludes the proof.

DEFINITION

5.6*.

We term

v

(

f

function of

(

LEMMA is at the

(

u

5.27.

same time

u

.....

)

on

^

,

the associate gauge

)

v^,....

.

Any symmetric gauge

function

the associate of its associate

(I)

*^T

(

(

v f

u ,

,

u

.....

v^,....

)

)

.

on

j

CROSSED UNITARY SPACES

V.

Proof. Denote by

N

there exists an

The

the associate of

^f

such that

u^

=

for

l>

last represents the value of the associate of

By Lemma

5.24

$N

(

must be equal

it

u

.... |f

t

U

N

For

.

"|p"

91

N

7

By

.

* or

xjf

u |f

(

(

IL,....

)

in

definition,

u

*

uw

)

to

= )

$(

u

on

K

|f

....,

U

.....

,

.

N

)

This concludes the proof.

Once a norm K,

(

*P

methods

may be

)

which

^

is defined

we

obtain a

in general will not be complete.

of " completing*

J^

(
)

,

that is,

normed

linear space

Details about the possible

imbedding

a Banach space

it in

found in a separate publication of the author. Here let us just outline

two methods for which the resulting two Banach spaces will

in the light of

the next chapter -- be closely connected with, and shed additional light on, the mutual relationship of the associate and conjugate space for

whenever oC (1)

the

is a unitarily

invariant crossnorm:

Cantor-Meray method

mental (Cauchy) sequences

of

[6, p. 106]

elements in

>2

(

by considering the funda-

,

$

)

anc* introducing

some

standard identifications, (2)

real

the "strong" method, by considering all those infinite sequences of

numbers

(

u,

,

u^,

....

-- having perhaps an infinite

)

number

of

non-zero

terms -- for which

.J!52.$ The

last definition

(

V

makes sense

.....

u^, 0,0,....

in the light of

carries over to symmetric gauge functions

)<.+..

Lemma

<J>

on

y^/

5.16,

which clearly

CROSSED UNITARY SPACES

V.

92

the smallest

The Cantor-Meray "completion" represents in

which

(

j

$

can be imbedded and

)

1

It is

"strong* completion.

and

(2)

furnish the

$(u |V when both

= )

1

"limiting* case for

21

(

=00

p

(1)

m

)

of all

,

if,

bounded sequences

l

|

<|>(

u

,,..., (

,

uj (

p. 11J

(2) will

for every sequence of

JUm^ we

[i

P

(u ,uv,....

p. 12^

.

However,

>

1

and in

)

f

,

fc

in the

,

c

while procedure

Clearly, procedure (1) and

and only

1

max

)

[l, p. ICl]

for

)^

furnishes the Banach space

ging towards (

p |u.)

when

for instance

(1)

that is,

=

(^(u^u^,....

procedure

many important cases procedures

true that in

and () furnish the space

(1)

clearly always included in the

is

same space. This happens ....

Banach space

,

*

)

(2)

& H sequences conver-

furnishes the Banach space

.

furnish the

numbers

u^,

o

(

same completed space

u., u^,....

.....

)

if

for which

)

have,

It is

not without interest to conclude this section on symmetric gauge

functions with the following simple theorem whose details

author's publication

Q^aJ

The conjugate space

v($

Meray closure

of

)

**iay

be found in the

:

of the

Cantor-Meray closure

characterized as the strong closure of

closure of

may

^( Tl7)

.

of

j^

(

^

)

may

be

Conversely, the strong

be interpreted as the conjugate space of the Cantor-

CROSSED UNITARY SPACES

V.

The class

7.

of unitarily invariant

we

In the following discussion

crossnorms on

fC

Q

v

shall derive an explicit representation

crossnorms

for all unitarily invariant

93

on

OL

O

9\,

pv

where Pv

,

is

any

unitary space, that is either a finite dimensional Euclidean space or a Hilbert

space.

We

shall also prove that the class of unitarily invariant

coincides with the class of uniform crossnorms.

crossnorms

OC

In section

1

of this chapter

OC(A)

norm and

DEFINITION of finite

For

.

this

was pointed

it

all

operators

most

shall use this representation

nition of a

Moreover, for these

sot'.

represented as the linear set of

accordingly

crossnorms

of the

reason we find

A norm

ol

is

A

on

may

Fv G> Pv

Ft/

be

We

of finite rank.

time and denote the crossnorm oC

later its associate in

5.7.

out that

it

terms

advisable to restate the defiof

operators of

any function

of

<X(A)

finite rank:

A

operators

rank satisfying the following conditions:

^

(i)

06(A)

(ii)

OC(cA)

(iii)

06(A

A norm

is a

-f

=

B)

JcJ

4

crossnorm

if

+

A

=

for any constant

cX/(A)

C*(A)

-r>

=

OL(A)

;

QJ,(B)

.

c

.

in addition to

(i), (ii), (iii) it

also satisfies the

following condition: (iv)

=

O(,(A)

Uniformity for (v)

o(/

OC(SAT)4

A crossnorm

is

clearly JlJS/il

A

for all operators

lllA/JI

means

,||TJ/j

that

Oi(A)

it

of

rank

1.

satisfies:

for any pair of operators

termed unitarily invariant

if it

S

,

T

satisfies the following

.

CROSSED UNITARY SPACES

V.

9^

condition: =

Ot(UAV*)

(v*)

& be

Let

A

operator

*fr

of finite rank.

AA

Then,

which

a^s(A)

is of finite

of its

By Lemma

The unitary invariance

unitary.

of OL

Suppose that for two operators with multiplicities of

V

a unitary

a

,

such that

$( where of

&^

*fr

abs(A)

We

.

a^,....

f

a

*7f

A

5.1,

,

and only .

^fc

=

Let

Uabs(A) =

of finite rank the

=

U

is

oC(abs(A))

proper values

are the same. Then there exists

abs(A)

=

finite

where

A

A

a.

<X,(A)

Hence,

.

Thus,

.

=

c6(abs(A))

06 (A)

depends only

define accordingly =

are

^

ot(A)

)

^e

,

point proper values with multiplicities

.

Thus a unitarily invariant crossnorm for all sequences of real

numbers

a^,

a

oC/

defines a function J)

,....

(

a

,

a

,....

which possess these

properties:

We

^

^ or

O)

ao

(2)

a,^a x ^

^

extend the function

only

(1),

9

a^a^. ...^

Od(A)

,

.

Consider an

implies

abs(A)

C^(A)

a^....

|t

^^7/

and

Vabs(A)V*

and therefore

Ot(abs(A))

on the

abs(A)

V

rank, Her mite an and

^

proper values (with multiplicities) are

stand for these proper values.

,

also of finite rank, Hermitean and definite

is

has a pure point spectrum; all its proper values are

number

U

for any pair of unitary operators

,

a given fixed unitarily invariant crossnorm.

Hence,

definite.

OC(A)

as follows:

on ly a

7/0

^

Let

defining

finite

number

of

i

*s

.

it

a,, a^,.,..

for all sequences for satisfy (1), and

which we assume

a^TTj....

be that

)

CROSSED UNITARY SPACES

V.

permutation of 'a |

,

V

.....

fact that

(T>

$(,. The that

must

it

Let a

,

a

A

abs(A)

=

By

$(VS

=

put

.....

)

relationships which can be obtained as follows:

We

be a fixed cnos.

21^ ac P^

S.JaJPj

Pc

define

A^A

Furthermore,

.

.

$(i.

a + b (

, (

a^

Hence,

.....

laj

,

|a,l

*"

}

=

a

a

....

(i)

$(aj

(ii)

<(

ca,, ca^....

(iii)

<J(

a|

f

, (

(E^j

/

and

=

|c

)

v ....

ca

to

,

a^

of

1

,

=

^

$(a

1

(

,a^

a |f

t

=

.

.....

b

to

same cnos =

...

....

)

)

a^,....

Thus>

'

>

,

b

,....

( <S/.)

we

^(e.V.V*

z',....

.....

for any constant +

$(b,,bv ....

)

denotes any permutation on the natural

= )

ot(P, 1

= )

1

.

get

.

implies clearly,

Finally, since ot is a crossnorm,

$(1,0,0

&fc

are the

a^a^,....

numbers.

(v)

2E.^

and

P^.

ca^,....

a^

=

Consequently,

.

,

A

and

$(W""

*

^ $(

)

)=

$(a,,at,.... =r

abs(A)

unless

b

a^f

The unitary invariance (iv)

2E

that is,

,....

a(A)

a^,....

>0

)

if

+b

, (

=

Let

(1)

respectively, always with the

....

b^,

Then,

~7/

generated by a unitarily invariant crossnorm implies

is

Applying the last equation to

where

e^T'

be any sequence of real numbers satisfying

*

and to

^

we

definition

point proper values (with multiplicities) of

OL(A)

*a f

=

c^,....

,

which

for

.....

,

>

some

satisfy

r^j

.....

Then,

jaj

,

satisfies (1) and (2).

.....

a^,

JaJ

95

Hence,

.

The preceding discussion may be summed up as follows:

c

)

LEMMA ates a (i)

CROSSED UNITARY SPACES

V.

96

--

A

5.28.

crossnorm Ok on

unitarily invariant

symmetric gauge function

(5

(

a,

,

gener-

satisfying conditions

)

a^,....

Pv

F\,

Definition 5.5' in the following manner:

(v) of

We choose

a

cnos

(

r4

For any sequence

,

)

of real

numbers

a., a

,.

satisfying (1) define

For

the so defined

whenever

a

i^

plicities) of

we have,

CI)

a-"7^-"

abs(A)

"/^O

represents the point proper values (with multi-

.

Conversely, we shall prove that every gauge-function satisfying conditions

(i)

-- (v) of Definition 5.5* can be derived in the indicated

finite

To achieve

crossnorm.

a unitarily invariant

dimensional case.

From

this, as

we

this

we

manner from

shall consider first the

shall see later, the proof can be

readily extended to the infinite dimensional case.

Our discussion found in [23,

A

p. 293,

will be

based on the following

Theorem

l]

.

The symbol

Lemma

t(A)

which may be

stands for the trace of

(Definition 5.3):

LEMMA U

fixed and

,

5.29.

V

,

The

maximum

of

are running over

all

^tfUAVB)

,

where

A

unitary operators on an

sional unitary space

Fv^

where

are the proper values (with multiplicities) of

a

,....,

a

is

,

B

are

n-dimen-

given by

abs(A)

CROSSED UNITARY SPACES

V.

b

while

denote the proper values (with multiplicities) of

b^^

,

, I

both monotonously ordered, that

Let space

n-tuples of real numbers

feJv^of

and

(i), (ii), (iv)

(v) of

n-dimensional unitary space

x

We

<|>

shall investigate the

X U

for unitary

sup

running over

is

V

,

x <S>

i

(

x

(

(

^

x f

+

stand for

that

& X

For any operator

all

numbers

of the

and

abs(UXV)

1

.

Since

Consequently,

.

)

S^ a f

are

a/v^

c

x.

fi xeci

t ^ie

"

.

,

proper values x -^

=

x^r * }

u

<J

,....,

x^)

*j Xj,

)

-

1

)

(

7/

A

.

"/^

*

little

a*,

=

1

a^x^ +

rnay be omitted.

X ^....

= )

(

a -

x^.

^x^-^-0

may

c

In fact,

and

x^

and do not decrease a.

x^ ^

....

f

and

consideration shows however that the re-

then interchanging

,

abs(A) and the

of

x

(

we do

-

2L^, a i x -

)

a^.

x^.

x^

be replaced by

whenever

)

<.

x

not affect

the change being

i;

^ x

x^

.

^

Thus, the .

.

A

posses the

abs(X)

$ (X)

abs(X)

=

dp (X)

"UV ^t(UXVA)

(

on

when

*2^t-t(XA)

operators for which

are subject to the restrictions

,....,

condition

"Y

xj

=

=

^

&f&

quirement

^

and

u^)

5.29, the last equals to

I

for a pair

.

put

x,,....,

4>00=l

x

,....,

,

^(UXV) max sup

sup

where the

(

the operators

,

7?,t(XA)

$CX) = /

Lemma

, f

we

,

sup

same proper values, we have,

By

u

denote the proper values (with multiplicities) of

x

,....,

and

is fixed

(

Definition 5.5.

Fy^

=

$(X) where

,

^ b^

b "#

For our immediate application we may assume

satisfies only the

and

^ a^

aj^

is,

abs(B)

be a symmetric gauge function on the n-dimensional linear

<J>

associate.

its

97

But even the

last

be omitted, because whenever

may

we do not

affect

SSI^ja^x^

x

CROSSED UNITARY SPACES

V.

98

x

(

,...., (

we are

Thus,

.

Definition 5.6 the last

We may sum

LEMMA and

\j/

X

and

(

v

=

Let

\Jf(

v t

A

1

,...., v^,)

t(XA)

=

(A

X*)

,

and

(

its

with

jel

=

1

.

unaltered, while

<^

a, ,....,

x

(

(

,....,

c.

where =

x^

the

1

.

By

a^)

be a symmetric gauge function

u^)

Then,

A

when

Ifc t(XA)

u

X

,....,

is fixed

is

*y

(A)

form:

be a symmetric gauge function

u^)

or

J

fv^,)

numerical value

Then,

assumed

is

its

in the following equivalent

associate.

t(XA)

|

to

with

sup

vary over

^(X)

=

(A

J

all

1

,

,

X)

/

operators (on the

assumes a maximum;

is

and

we argue

sup |t(XA)J

,....,

Lemma

That both last

Proof.

^T^ at x

operators (on the n-dimensional unitary space Ft^

all

Let

5.30*.

numerical value

u

(

n-dimensional unitary space its

-x^

readily seen do not decrease

sup

assumes a maximum, and

,

is fixed

^f (

associate.

its

v^)

sup

when

equals

<J>

shall use the last

LEMMA and

is

by

x^.

up our discussion in the following:

5.30.

,....,

(X)

We

and as

1

then replacing

,

really dealing with

sup

runs through

with

=

x^)

<

are subject to the sole restriction

x^

,.,..,

x^

are equal, follows from the fact that

sup's

$ (X)

=

as follows:

Replacing

X

^R^{6 t(XA)

as

<J

To prove

fchat

sup

"2t(XA)

denote any complex number

Let

$X

by )

.

<J(X*)

,

<

varies

($X)

=

$(X)

assumes

a

remains

maximum

)

CROSSED UNITARY SPACES

V.

equal to

|t(XA)|

LEMMA

The rest

.

Let

5.31.

(p

,....,

For an operator

(Definition 5.5).

This concludes the proof.

is clear.

u

(

u

99

be a symmetric gauge function

)

A

on an n-dimensional unitary space

(fi (

a

K^

put =

(> (A)

where

a

,

,

a^)

,

, r

,

denote the proper values (with multiplicities) of

a^

abs(A)

.

Then,

<&

(

(A)

^0

(i)

CJi

(ii)

i(cA)

(iii)

<|>(A + B)

(iv)

(v)

<(A)

Proof,

(i).

aj,...., (ii).

=

^

HI

A

,

1)1

=

a =

Jc I*"

represent the proper values of

(iii).

Let

'XlT

,....,

v^)

.

,....,

+ B)

sup

1(A,X)|

=

sup

TtfW +

a^=

,

^A

,

abs(cA)

v

.

<J

(

c

U,V rank

.

<

1

Now,

=

that is,

/TA

numbers

the

.

=

(A) =

and

Icja^

,

A

=

teja^

Hence,

A

be the associate of

)

5.24,

By Lemma

$ z (A

Y00*i (iv). This

v

By Lemma

(Definition 5.6).

"ty( v

(

=

....

is of

is clear.

,

t

(cA^(cA)

A

when

=

.

for unitary

$ (A)

implies

Since

<(B)

f-

$ (A) ^

That

a^)

(A)

<

A

implies for any constant

|c| (^ (A)

=

="

=

$(A)

;

,

Uj,

u

is

)

(

u

,....,

u.)

also the associate of

5.30',

<

(A +

B

sup

I(B,X)(

,

X)

I

=

A(A)

+

<5(B)

.

YCx)*i is true

since

(UAV*)*(UAV*)

=

V/FAV*

and

A^A

have

0.

00

V.

the

same proper values.

1

with

and

rank

is of

,
Cp*

its

A

If

(v).

CROSSED UNITARY SPACES

Then,

.

(

&^>

(f

only positive proper value is

which clearly represents the bound

REMARK norm on for

^

that is,

,

abs(A)

ll(p||)irp||

of

5.31,

$

(A)

1

^

$

,

1

,

(B) 4-

c ,....,

c^

abs(B)

,

$(

We

=

,

,

a,,....,

t^rp

!|

f

II*

^

of

Thus,

.

.

This concludes the proof.

is a unitarily invariant

<j)(A)

cross-

is naturally the triangle inequality

+

$> (B)

1

=

)

=

Therefore,

.

p

;

we may express

Since this

.

,

^

q

,

is

equivalent to

P + q

=

a^

b

1 ,

the following:

and

a

let

|t

a^,....,

;

,

f

b^,...., b^

denote the point proper values (with multiplicities) of

and

+

abs(pA

qB)

gague function

A

p^0,q-#0,p + q=l,

Let ,

(

^ $ (A)

(A + B)

<J>

+ qB)

implies

c

(J>

f C(>^

The interesting part

Fy^ fVw

saying that

and

By Lemma

5.7.

We assume A

the proof is trivial.

,

a^)

^

1

^(

,

b

respectively.

Then, for any symmetric

(i)--(iv) of Definition 5.5, the inequal-

,....,

{

bj 4

1

,

imply

$(

c

...... f

c,J<

1

are just ready to extend our result to operators on a Hilbert space

\

and prove the desired converse of

LEMMA

5.32.

Let

<J>

(

a^

,

Lemma a

5.28,

.....

)

denote a symmetric gauge function

on Vw satisfying conditions (i)--(v) of Definition =

where of

a,

abs(A)

^ a^-^

.....

^

a,,

Then

5.5*.

ax,....

the equality

)

denote the point proper values (with multiplicities)

defines a unitarily invariant cr os snorm ot on

fvO

PC

.

.

CROSSED UNITARY SPACES

V.

Proof.

Clearly,

Lemma

Furthermore, by subspace for

then

,

which the ranges

A

operators

A

fv

of

A"!&

,

norm

A

is defined for all

)

5.31,

Q(A)

of both

B

,

*"

whenever is a

A

A

B

,

.

Clearly,

B

,

of all

Condition (v) for

Q

if

A

is restricted to

operators on (iv) for

^

f\/

Accordingly,

let

f(,

3\, be spanned by the ranges of

*T

dimensional and

Jv is finite

is

c^(A)

a

which contain both (arbitrarily given!) of a

norm never

involve

This implies that the defined above 06 FN/

a

is

more

norm

of finite rank.

implies the unitary invar iance of 06 1 .

and finally

This concludes the proof. the rest of this

stand for an n-dimensional unitary space or a Hilbert space. K/

stand for the set of n-tuples of real numbers, or for the

linear space of infinite sequences of real

a finite

number

crossnorms on

fCPv

numbers having only

non-zero terms.

THEOREM and the class

Proof..

of

5.1.

The class

of unitarily invariant

symmetric gauge functions generate each other.

The proof follows from Lemmas

The discussion which follows throws

.

those operators

Now, given any two

are in cJv

theorem which follows, as well as throughout

Chapter, let

of

A

01

any fixed finite dimensional

is

assures for QU the "cross-property*

In the

operators of finite rank on

The defining properties

than two operators at a time.

on the set

and

H/

of finite rank, let

,

for a family of operators

operators

norm

"""Jfr

B

,

o&(

1

5.28 and 5.32.

additional light on

Theorem

proving that unitarily invariant and uniform crossnorms coincides.

5.1

by

1

02

CROSSED UNITARY SPACES

V.

LEMMA

A

Let

5.33.

any operator on fi

Let

.

af

^

1

a^

"Jfr

^

and

b.

denote the point proper values (with multiplicities) of

Proof.

b^

Then,

respectively.

We

X

denote an operator of finite rank and

a^

|)|x||[

^ bx *^

^

....

and

abs(A)

be

abs(XA)

.

use a well-known theorem of CourantfZ,

p. 2?3

It is

.

s

stated there for finite dimensional Euclidean spaces, but

it is

clear that

applies equally to Hilbert spaces, always assuming that the operator

question has a pure point spectrum (with multiplicities)

c

^

c

it

K

^....

in

^

.

This theorem states:

min

=

c^

where

max

(Kf

,

)

f)

the right side in the last and following three equalities should be read

Let *HC be an i-1 dimensional linear manifold. Define

as follows:

max

as the f

(

(Kf

orthogonal to

for all

i-1

over the set of

f)

,

7ft{,

.

We

with

f's

min

consider then the

dimensional

Now A A

all

7Ft>

II

f

(I

of all

=

1

and

numbers

Since,

ac

^

and

(A*Af a^

Similarly, of

we form

abs(XA)

.

=

min ,

f)

min

(

=

(

max

(li^Af

Af

Jl*"

max

|]

ll

,

Af

,

we

)

f)

have, )

J|

the point proper values (with multiplicities)

Then, bc =

min

VfQ

a^-^a^ ^...

Hence, =

1tv(

-

has a pure point spectrum (with multiplicities)

a^

*K( 7fC)

(

max

JJ

XAf

jl

)

.

b f

CROSSED UNITARY SPACES

V.

Clearly,

||

XAf

|j

<

JJJxJI|

I|A

b 4|J|x||| a^

Hence,

.

I)

1

03

This concludes

.

the proof.

LEMMA

Let

5.34.

operator of finite rank.

and therefore =

Ot(A*)

A*

=

LEMMA

Let

y

aj

a%

^

A

A

Proof. Let

For

o(/

5.33,

b

we form

the

U

.

Thus,

ot(A)

=

such that

A

=

Uabs(A)

and

oC(abs(A))

crossnorm

is

OC/

and

b,

a^

.

^ b^>

....

abs(A)

and

Hence,

b

abs(XA)

=

symmetric gauge function

any operator.

denote the point proper

~%

c

uniform.

X

be an operator of finite rank and

j/|x|/J

t

.

there exists a unitary

values (with multiplicities) of

Lemma

Ot(A*)

unitarily invariant

^

....

=

an

This concludes the proof.

.

5.35.

5.1,

Ot(A)

abs(A)U*

Ofc(abs(A))

A

be a unitarily invariant crossnorm, and

Then,

By Lemma

Proof.

ot,

Blx/|/

$

respectively.

p^

where

as stated in

By

04-p^l

Lemma

5.28.

Then, Ol(XA)

=

)=

$(b,,ba.....

The above inequality sign

is a

<$( BIXIHp.a,

consequence of

carries over to symmetric gauge functions on 06

Now, by

Lemma

for any operator

5.10,

Y

,

(XA)

^

Ot(A)

=

we have:

III

X

HI

OCC)

<X(A) .

Lemma

W

.

IIIXIII

px a% ,....

= )

5.16 which clearly

Thus,

.

Thus, by what we have shown already,

CROSSED UNITARY SPACES

V.

=

UlY*lli

ot( (AY)*) =

OCXA*)

fl/Y/H

*

<X,(Y*A*)

c*(A)

^

.

Therefore,

06(YAX) 4,

III

Y

I//

06(AX)
ll/Y/JI

OC,

(A)

.

This concludes the proof.

LEMMA

III

We

U

Let

Proof.

A

5.36.

U

uniform crossnorm

and

= l||

III

is unitarily invariant.

denote any two unitary operators. =

III

1

III

;

V HI

=

/|/V*/|/

Then, =

1

.

have,

Ot(UAV*)

On

U*

V

CO-

lliU/H

oO (A) JriV*W

=

06 (A)

.

the other hand

<X(A) I/iu

K J|j

=

Qt(UAV^)

lliv

Thus,

This concludes the proof.

We

combine Lemmata

5.28, 5.32, 5.35 and 5.36 in the following

statement:

THEOREM

5.2.

The class

of

uniform crossnorms on

cides with the class of unitarily invariant crossnorms on last class and the class of all

symmetric gauge functions

satisfying conditions (i)--(v) generate each other.

pv

fv

P\^Q *J>

(

P\s

a^,

,

.

a

coin-

The .....

)

CROSSED UNITARY SPACES

V.

THEOREM operators on

of finite 0^-norrn, that is,

ft,

(Definition 4.1)

and only

if,

This

Proof.

For a given crossnorm

5.3.

THEOREM

is a

(

f(,

the

Banach space

forms an

&^%,)

of all

ideal

O*/is unitarily invariant.

if,

consequence

The bound

5.4.

,

105

of

Theorems

of

>(A)

4.2 and 5.2.

A

an operator

represents the

least unitarily invariant, consequently uniform crossnorm.

Let 00 be a unitarily invariant crossnorm. By

Proof.

form a corresponding

may

^)

By Lemma

.

5.5,

be represented uniquely in the canonical form 2l'e =

>(A)

III

A Id

max

= I

by

Lemma

On

5.3.

max ^C4->v

$13"*,

|

5.28,

of finite rank

^s

a
bound

a^

the other hand, Lerr\rna 5.17 gives,

^ $( *

a^

A

an operator

Lemma

a

,

a

'

=

.....

)

*

<X(A)

.

Thus, for every unitarily invariant crossnorm 06 we have clearly unitarily invariant.

OC^^

.

But

7^

is

This concludes the proof.

Although the unitarily invariant (uniform) crossnorms form the significant class of

norms

norms which are

norm which

is

it is

not without interest to construct examples of cross-

not uniform.

not

From Theorem

"^"Xis not uniform.

We

5.4

it

follows that any cross-

shall construct such

crossnorms

in the latter part of this chapter.

We

conclude this section with the proof that for every unitarily invariant

crossnorm

Ot

,

we

have,

6C

~ 06

.

106

DEFINITION on

Let

5.8.

(A

|

X

associate

A

its

X)|

X

Ot<

on Fv of of

LEMMA

^

,

the

OCs

^

c

finite rank, let

be the

O(/(A)

OUX)

on Fv of

a consequence of

operators

on fv of

on Fv of finite rank (or what amounts to the same

thing, for all operators

It is

A

satisfying the inequality

c

for all operators

be a unitarily invariant crossnorm

OC/(X)

For an operator

Pv

fv

least constant

and

CROSSED UNITARY SPACES

V.

Remark

rank with

5.2, that

finite rank, coincides

when applied

,

finite

to

fv

Pv

Ot/(A)

=

O^(X)

1

)

as stated above for

with Definition 2.2 of the .

Pv O Pv

Let 06 be a unitarily invariant crossnorm on

5.37.

symmetric gauge function

it

.

generates (By

Lemma

5.28).

Then,

associate d? generates the associate gauge function "^"(Definition 5.6*)

Proof.

Lemma

5.30* proves this

when ^/

is a finite

dimensional

Euclidean space. Here we extend the proof to the case when fv space.

We show

V"

a l* a

So

that for an operator

A

on fC of

finite rank,

let

A

be an operator of finite rank on Ft and

operator on T7t

may

a^ CP.^^,

be by

Lemma

5.5,

*Ylt

of

Hilbert

we have

being the point proper values (with multiplicities) of

m-dimensional linear manifold generated by the ranges

form ZT

is a

,

abs(A)

.

the finite say

A

jfcfe

and

PL

.

An

represented uniquely in the canonical

which clearly also defines uniquely an operator on Fw

Thus, the linear space of operators on *nt

may

be identified with the linear

.

CROSSED UNITARY SPACES

V.

space of of

A*

is

We

included in 'fit

&J

put

x,,.....

denote the associate of

T]/JX)

=

X

operators

.

<(

=

xj

Then clearly,

X

on |

.

*X/C

(A

,

hence X)|

,

c

rf)

is

=

|(A

,

X)|

5.30',

^

.

c

\jjA)

$>^X)

for

X

on fv which

Thus, the requirement for this

obviously weaker then the one for

in Definition 5.8,

CC(A)

Consequently,

On

always find a ranges of

X

the other hand for an operator finite

A

,

m+k -dimensional

say

A^

X

,

X*

,

Such an inequality holds for any the converse,

THEOREM

X

^ *(A)

ot'(A)

on fv

*

finite

linear manifold containing the

.

on

rf/

of finite rank.

This furnishes

This concludes the proof.

Every unitariiy invariant crossnorm

5.5.

rank we can

Consequently,

.

ot

on

a

satisfies the condition,

Proof.

(Lemma

Let

5.28).

^>

OC = OC

.

be the symmetric gauge function generated by OO

Then, by

Lemma

5.37,

Ot'

is

introduced identification,

for all operators

(X)

and

(X)

(f)

^f^

and let

)

By Lemma

in the light of the

$

.....

.

d) (X)

on Y/l

be identified with the operators on Tft

constant ^f/^t,(A)

w

x

......

Xj

satisfying the inequality

also the inequality

may

.

^f^

c

0?

on fv whose range together with the range

for operators

^(X)

the least constant all

A

those operators

all

1

generates the associate of

$

,

108

CROSSED UNITARY SPACES

V.

and again #" generates the associate of the associate with

^

by

same

the

8.

Lemma

Thus,

The space

,

which coincides

must coincide with OC since they generate

(

V

OC

-norm

Moreover, the last

By 06

if

(Definition 3.2)

and only

always represents

sup

||

A

Definition 3.2, an operator on H/

-norm

if

An

operator

A

if,

_

sup

Proof.

PC0&

Let Ol be a crossnorm on

5.38 .

is of finite

is of finite

o(/'

^

Th^ s concludes the proof.

^

LEMMA on F^

5.27.

of

and only

if,

.

|J

,

that is,

from

F\/

into f\

c

such

there exists a finite constant

that

for every operator

denoted by

is

||

A

ZTTaiH?^ J)^

.

Now

By Lemma

5.12 and 5.15,

<Elut( A(Pc

^

H

A

'

by

t(AX)

Lemma

H^ of Definition 3.2.

if

frequent applications

n

X

FvOpv =

2ETJ.,

the least of such constants

;

^

sZ^fAtfl.

5.13 (iv).

,

Jf|.

then

AX

=ZT

Hence also

,

eft)

Thus, above

sup

(

cfc

A(^.

t(XA)

=

coincides with

This concludes the proof.

The theorem which follows its

i

we

is a

restate

it

particular case of

here

in the

Theorem

3.1.

Due

to

language of unitary spaces.

THEOREM A

operator

5.6.

CROSSED UNITARY SPACES

Let

$

on Fv of

fv

(

!* iFv

defined by

means

XC

for

t(XA)

of (i),

Then, there exists an

)*

F(,0R

.

A

Conversely, whenever for an operator is

109

O(,-norm, satisfying the following conditions:

= Illtf/JI

JAIL

(ii)

L

finite

=

<^(X)

(i)

V.

V ^

we have

(

on fv of finite ok -norm

fCfl^Fv

)

and

(ii)

also holds.

denote an additive and bounded functional on

Proof. Let 3 Clearly,

and

Thus, holding (D fixed and varying the

complex conjugate

of an additive

Lemma

by a well-known

y(
such that

of

argument

III

A

(i)

III

(f

,

Of

)

.

=

holds.

4-l|Aa

Z r; ^ t

A

t

(

<6fjL)

holds.

=

by

for all

Alp (p

f

in

Op =

^s

)

Hence

.

=2:7,^^.^.) It is

and sup

^ fv

.

=

it

Thus,

*p

,

t

.

t(XA)

(

is clearly additive.

ot(x) (ii>

A

Define

^

=

*

Thus

fy ( Cp^rf

F. Riesz there exists a unique element, term '

=

that

have2;^; (A(pc ,^) ZTVfc*^* proof of Lemma 5.38. Consequently,

in the

Thus

X

for

we see

and bounded functional on Fv

(A

Now

f>Lf

t(xA)

also bounded since by

by an

.

Lemma

3.2,

v

MO

CROSSED UNITARY SPACES

-

Conversely, Let and

is additive

II

4^1

pv

by means of

bound on

its

represents

be extended uniquely without changing fC

$

<-foo. Define

11

OL

A ((^

(|

A

its

bound

fvC^ft

.

Then,

(i).

V

Clearly,

can

an additive functional on

to

This concludes the proof.

.

*

THEOREM

For any crossnorm 06

5.7.

Banach space

the

pv

(

X

may

be interpreted as the space of

all

operators of

<8^

oO-norm on Pv

finite

-M

Pv

)

,

where Q***KQ* represents the

norm

of

The proof

Proof.

A is

*&>??

(

We

)*

consider

of operators as the

norm

operators

(sc)

By Remark

.

that goes with

X

tpa^

(X

,

5.1,

of finite rank, that is on

,

and

Lemma

5.38.

cross-space

Remark =

<pa
Thus the normed linear space

The characterization a special significance.

(

space on which

fv

O

<$(X) fv

.

=

(X)

is also a

norm

Moreover,

Cf

(X

for all

has also

5.2,

f

l^/Ogfv

of the

is a linear

(sc)

(Definition 5.2), with

Y)

In particular,

it.

the cross-property, since by (

5.6

.

there is defined an inner product as the

Theorem

a consequence of

The Schmidt-class

9.

^(x)

.

is

.

to)*"

included in

crossnorm

<

=

(sc)

Iflf Hop//

.

.

which follows attributes

it

,

X)

CROSSED UNITARY SPACES

V.

DEFINITION =

Ot(X)

LEMMA

5.39.

oJ(X)

We

Proof. i

A

5.9.

crossnorm Ot

termed "self-associate"

X

identically (for all operators

(5

if

of finite rank).

unique self-associate crossnorm.

is the

shall prove first that

<$

*

'

<>

.

A

Let

X

be a fixed and

variable operator of finite rank. Schwarz's inequality (Remark 5.1) gives,

|(A,X)|

tf(A) (>

(A)

=

)*

(A, A)

=

'(A)

for all

6(A)
<,

^

& (A)

Thus, Definition 5.8 gives (

is

1 1 1

that is,

in

On

.

.hence

tf\A) <S(A) ,

^(A)

X

FvOPC

.

the other hand obviously

<S(A)

^

'(A)

Thus,

.

a self-associate crossnorm in the sense

($

is

we

first

of Definition 5.9.

To prove

the uniqueness

remark

that the definition of the

associate for a given crossnorm implies

% 6(A)

(

A

for any operator

get

^(A)

^

<X(A)

therefore also

Thus,

Qt (A)

OL(A) =

)

=

(A.

of finite rank.

Applying

.

^

<^(A)

^(A)

A)*

Assuming

Lemma

2.2,

OC(A)ou'(A) thus,

we

get

since by assumption

ot

srot'

identically,

ot'(A) ^.

oi~ot and

for all operators of finite rank.

and

<S'(A) <3 **

we

o

.

This concludes the

proof.

THEOREM

5.8.

class of operators on

Proof.

normed

We

(

Fi/

/v^ft

)

may

be considered as the Schmidt-

.

have pointed out before (Remark

linear space.

By Theorem

5.1) that

5.7 it is sufficient to

(sc)

is a

show that an operator

A

112

CROSSED UNITARY SPACES

V.

is in

and only

if

(sc)

if, it

is of finite

-norm, that

<

is, if

and only

if,

and also

A B,

II

Let

A

H

ff (A)

A ^

Since

A fv 1^ >

.

for which

be an operator JL

stand for a cnos. 217.1

=

l|

A

H^

< *oo,

for sufficiently large

,

Consider ZT,

^

We

and

(

(

we have,

n

have,

and

Z,

H

A ?, f =2:^,

(A
A^,

,

)

Consequently, X )*4 lAll^ (2:^ A fJ|

A C

that is,

On

and

(sc)

By Schwarz's

rank.

=

|t(XA)|

Lemma

Thus,

implies

Conversely,

*>

|(A,X*)j fl

and

(sc) if

A

A

||

<S(A)

^

tf

=

flAB^ ^F (sc)

of

A

||

tf

^

our proof

II

that is,

,

A

f/

A

**(A)

tf(A)

G?(A)

,

is,

<

||

A

.

||^

be any operator of finite

.

=

tf(A)

(?(X)

Therefore,

||

A

.

fl^

<

-^

<

.

is

defined and finite.

Then, the

Schwarz's inequality implies

is of finite

.

g(X^

(?(A)

second half of the above proof, that ||

X

and

(sc)

)

inequality:

5.38 gives

A ^

(2I c liAyJ|

A 6

the other hand, let

4

1'

=

rf(A)

n=l,2 .....

for

rf-norm, and thus, by the first part

This concludes the proof.

CROSSED UNITARY SPACES

V.

&

-

THEOREM the

F/

have

=

Fv^ Pv

(

)

f Cv

;

it

3

represents

Euclidean space or a Hilbert space according to whether

n*^- dimensional is

We

5.9.

1 1

an n-dimensional Euclidean space or a Hilbert space.

Proof.

It is

when Pv

clearly sufficient to consider the case

a

is

Hilbert space.

A

Let

(sc)

.

Lemma

completely continuous. By

lim a c

=

,

and both

Thus, for a given

<.

(

"7-0

-*:?,

llA

"generally" known that an operator in

It is

.

and

)

we can

,

a.
(

=

ft<3rf&)*

Now

/V^/P^

the separability of

Lemma

see this

we

normed

linear space

for a cnos

a cnos in

Appendix

recall

(

*fo

)

Fv$fejv

It is

=

II

= rf

(sc)

in that

f=v

&

,

A

2L c a^^^cft where

=

ar ^ nos.

N

for

that

tf

5.8),

which

hence by Theorem 5.8

(9^^. (Theorem

of finite

of finite rank.

Thus,

3.6).

Fv implies the separability of

fv

^Pv

To-

.

r

which states that for any crossnorm ok the

the operators

fvti^pv

<.>

cp.

are separable. Moreover, ;

i

,

j

=

1,2,....

form

This concludes the proof.

may

sequence of operations,

,

(Z^t)*
not without interest to add (as follows

II)

a c >0

Clearly (Theorem

norm by operators

K/O^Fv and hence

in

.

2.4,

(rfc)

find an

This proves that every operator in

tf-norm can be approximated

5.5,

is

(sc)

from

a construction in

be obtained from any crossnorm by

means

of taking the associate, using the arithmetic

of a

mean

of

two crossnorms and taking the limit of a monotonic sequence of crossnorms.

The trace-class as the cross-space

10.

We and

The symbol

LEMMA

The crossnorms tf

a-

^

By Remark

V (A)

^

inequality,

m(A)

5.5,

A

are

a^*s

Lemma

(by

and the

"7

r>f>

r^

f

,

.....

.....

,

r^ to

,

pj/^,

,

a%

,

a

,

m(A)

=

The sum on

the

Thus,

crossnorm,

=

m(A)

)f f

is,

*"

*f

(A)

.

^

^ (A)

2.

i

(

absfAjrft

^

m(A)

is

r>pj

To prove

=

Hence,

the converse

form nos. We have,

^Js

a^

.

)

we see

that

,

a^

,

.....

,

a^,

,

.....

=

clearly

corre-

Therefore,

Consequently,

cfO

m(A)

,

AJ A

corresponding to the proper values

ZL

t

(

a

^(A)

c>

ffc

for all

=

^Zi a

C,

since by definition,

and therefore,

^ (A)

.

canonical form

are proper vectors of

abs(A)

,

and

s

.....

a^,

extreme right

^G pC

27..<'14
,

ofj^, cfa

a^, 0,0,....

......

crossnorm on

a

5.5) in the

=

they are also proper vectors of f

rank

of finite

^(A)

(Definition 5.4).

t(abs(A))

of finite rank.

*fV

a cnos

sponding to the proper values

a

for the greatest

.

=

A*A Extending

is

A

for all operators

we write

"where the

rfc,

A

f%/ (By fy

symbols

are associate with each other, that

A

*f,

=

)

.

Proof.

m(A)

an d

^

assumed

stands for

m(A)

For any operator

5.40.

fyflffcfv

(

of finite rank, the

stand for its bound and value

(A)

^J

A

recall that for an operator

respectively.

and

CROSSED UNITARY SPACES

V.

114

A

in

CROSSED UNITARY SPACES

V.

remains

It

we have

7*(

=

*^ (A)

Thus,

That in

Theorem

We

"^

15

*jj

(A)

^(A)

* "

2.5.

.

Since

tinuous;

,

and characterize

^ -^Y

its first

The linear space its

represents

its

of all

X

Clearly, every operator

its

norm.

fK/ is

A

let

5L

c

well as the

a

bound.

of finite

To complete our proof if

completely continuous

all

completely continuous operators,

Hence,

completely continuous

Banach space

the

rank

is

Fu0^Pv

where

it is

wiiere

at

form nos. By

>

% A Cv

represent the

the bound of an operator

sufficient to

and only

fv

completely con-

if, it

show that an

can be approximated

finite rank.

represent a completely continuous operator. By

fr
^*s

.

and second conjugate space.

norm, furnishes

bound by a sequence of operators of

=

crossnorm, we have also

linear space of operators of finite rank,

operator on

So

is a

holds even for general Banach spaces has been proven

5.10.

7i(X)

stands for

A

*X

.

with the bound of an operator as

in

gives,

This concludes the proof.

THEOREM

Proof.

5.3,

^A

definition of the associate

are about to consider the Banach space of

operators on FO

normed

The

1

Lemma. By Lemma

of our

5

>'(S7*,^C

=

T?(A)

=

fte)

2Tc^,
7f(A)

that is,

prove the last statement

to

1 1

,

Lemma

Urn a^ 5.3, the

=

,

Lemma

and both the

bound of

A

-

5.5,

<.f s as

< Slcl, a c

ft^;

.

116

is

CROSSED UNITARY SPACES

V.

given by

sup a^

lim

Since

a. =

,

the last

.

number obviously approaches

* oo

n

as

Conversely, the limit of a sequence of completely continuous (hence also of finite rank) operators convergent in bound is also a completely con-

tinuous operator by

Q,

The trace-class

Remark

5.5,

96]

has been defined before (Definition

(tc)

5.11.

The Banach space

gate space of the space of

all

which ZI^( (^A)*-(, (.

Proof.

which

is

H

By Theorem

A

^7l

is, the

< +

o*

space of

all

for a cnos

= 11^

all

crossnorm.

*.)

may

A

be inter-

on Pt for

and where the last

,

norm

of

A

prove that the trace-class

5.7, it is sufficient to

A

those operators

By

that is, the conju-

)

operators (

5.3).

independent on the chosen cnos represents the

represents the space of (i)

)

F6

(

is a

m(A)

completely continuous operators,

preted as the trace-class, that

infinite sunn

This concludes the proof.

.

forms a linear space on which

(tc)

THEOREM

p.

for which

sup _

and (ii)

||

A| A

We assume finite.

whose abs(A)

=

A ^

5.1 (ii), there exists a partial

initial set is the

which

.

first that for an operator

By Lemma

for

m(A)

the

number

||

isometric operator

A

is

|L

W

closed linear manifold determined by the range of

abs(A)

the closure of the range of

=

W*A

abs(A)

Let

.

.

We

^p.

,

ry "*

extend

,

op3

denote a cnos in

,....

*

it to

rfi

,

ry

,

^

,

,& ,CO ,G) t

,

CROSSED UNITARY SPACES

V.

a cnos in fv

which we shall denote by

Clearly, the

W/^

Lemma

We may

.

form an orthonormal set while

*s

also assume ^> =

WcO.

we have,

p

^(Zj^,

^ft

tf[.

c-

^

By

o

every natural

5.3, for

<)

(

1 1

)

=

*

Therefore,

.

2T,(A
The

)

last inequality holds for

=

m(A)

^

A

therefore also

On

w
,

This implies, and

||

A 1)^

A

II

=

m(XA)

^

||

m(A)

X

^

.

(tc)

IJAj^

^

part.

This concludes the proof.

-f<

COROLLARY relative to the

Proof.

LEMMA spectrum. plicities) of

If

,

a,

,

abs(A)

(g.) ^flA/^. Lemma 5.14 (vi) ,

} lj

A

(X)

JJ^

m(A)

<

and

(v) gives,

.

oo

implies

A 6.

Lemma

The second

and consequently

of

II

abs(A)

5.11,

^

(tc)

(tc)

and

half of the above proof gives,

A 1^

=

operators

m(A)

A

is

by the first

complete

norm.

is a

5.41.

Thus,

The trace-class

5.1.

m(A)

This

).

f

=

implies by

defined and finite.

<

and

abs(A)

.

rn(A)

m(A)

(abs(A)

)||xl||m(A)

thus

is

..... Thus,

of finite rank,

rn(A)

A ^

Conversely,

2^

=

t(abs(A))

^

1,2,

and

(tc)

the other hand for any |t(XA)l

=

p

consequence

For

A

a^,.... ,

then

5-

(tc)

of

Theorem

abs(A)

,

5.11.

possesses a pure point

denote the point proper values (with multi-

m(A)

=

2

L

a^

<

* o

t

7

118

CROSSED UNITARY SPACES

V.

(abs(A))

b

ator in

A

Let

Proof.

(sc)

(tc)

Now

.

By Lemma

.

it is

5.11 this is equivalent to

well-known that every definite Hermitean oper-

possesses a pure point spectrum,

(sc)

Let

sisting of proper vectors of this operator.

(abs(A))

abs(A)

Each

.

.

rj*

is

a proper vector for

a^ be the proper value

Let

proper vector <^

Clearly,

.

of

there exists a cnos con-

i.e.,

be

(rj^)

siflch

a cnos for

hence also for

(abs(A))

corresponding

abs(A)

to the

we have

This concludes the proof.

LEMMA in

relative to the

(tc)

Proof. proof.

The set

5.42.

For an

We form replacing

A

Let fc

>0

(tc)

.

B

use

C

,

finite

X

rank

is

dense

5.41 and the notation in its

,

which we obtain from

^

i

unchanged for

^

such that 3E > p a c

p

for

a^ by a^

Lemma

find a finite

and

proper Values

leaving its proper values

We

.

we can

operators of

norm.

m(X)

two operators its

of all

p

i^ p

also by leaving the corresponding proper vectors

,

by

abs(A)

or for

i

^

p

,

or for

i

^

p

,

cR unchanged

.

by and

in all cases.

Then, abs(A)

B with

has

abs(A)

.

B

and

C

The above properties abs(C)

and therefore,

B

rank and both

finite

=

=

+

C

.

are Hermitean and definite along

of

C

imply

(C*fc)*

=

(0*

=

C

CROSSED UNITARY SPACES

V.

e

C

Thus,

By Lemma

A

=

WB

+

and

m(C)

5.1 (i),

A

(tc)

WC

It is

.

m(A

-

= =

Wabs(A)

=

,

WB

clear that

WB)

4

t(C)

.

II

W

III

=

Dlwl||

1

.

Therefore,

B

rank along with

m(C)

^

,

e

This concludes the proof.

THEOREM Proof.

5.12.

The trace-class may be also interpreted as

By Corollary

relative to the

norm

5.1, the trace-class of

m(A)

.

In

it

rank form a dense linear subspace. since by

Lemma

finite rank.

5.40,

we have

*fi

by

Lemma

operators

is

complete

5.42, the operators of finite

K/0

This subspace coincides with

(A)

=

for all operators

m(A)

^(/

A

of

Consequently, the trace-class coincides with the smallest

Banach space

in

which

Ft/' R/ 9

can be imbedded, that

is,

with

Pv

f(s if

This concludes the proof.

THEOREMS. 13. Proof.

This

THEOREM space of

all

is

(

fv

^ Ft )*

a consequence of

5.14.

operators on

(

fC

9\/

3LJ& )*

where

the

=

Pv^'K/.

Theorems

ma X norm

5.11 and 5.12.

be interpreted as the Banach of an operator is given

bound.

Proof.

This

is

a special case of

9

.

is of finite

m(WC)^

1 1

Theorem

3.2.

by

its

and

120

CROSSED UNITARY SPACES

V.

THEOREM

The linear space

5.15.

of all

operators on fv where the bound of an operator furnishes a Banach space C(5

A

on

tf,

orthonormal set

,

(

for which

\fo

)

Its first

21(,(

the last

;

is the

(jfA)Q

sum which

,

is

complete orthonormal set represents the norm conjugate space

*^?

^^ may

of

is

considered as

conjugate space

The trace-class

preted as the trace-class. ators

.

completely continuous

may

<

Banach space

<+

its

oo

f

or a complete

of

A

in

^^

The second

.

be interpreted as the Banach space of

Proof.

The proof

COROLLARY Proof.

is a

5.2.

Fv

consequence

fv

By Theorem

?\j $9^

(

GTv

is a

^'^

2.5,

oper-

independent on the chosen

Moreover, *4J may be characterized as be interpreted as

be inter-

of all

operators on fv where again the bound of an operator represents

^S may

norm,

.

Fc>L^/

=

fv

)

of

fiL

Theorems

its

all

norm.

while the space

,

K/

,

and finally

5.10, 5.11, 5.12 and 5.14.

proper subspace

of

ftfC

(

)*"

Hence, Theorems 5.10 and 5.14

furnish the desired proof.

We

conclude this section with a few words about the inclusion f

DEFINITION a natural

where

the

5.10.

r

an y "limited" crossnorm

Let OL denote a crossnorm on

number. For^ an operator

sup

is

A

of finite

taken over the set of

all

Ob

f\/Q

.

f\/

and

p

rank we define

operators

X

of

rank

^

p

.

CROSSED UNITARY SPACES

V.

LEMMA

All

(iii)

Ot,

Ct-

*

"^

>

OC-

^ p

of (i)--(iv) is presented in

,X^ p>

5.44.

A

variable operator

X

X^

well as the

"^

.

implies

|5

Let

Proof.

Since

CC

OCp.

an immediate consequence

LEMMA

where

.

The proof

Proof. (v) is

are reflexive crossnorms.

Qfc^

lim

(v)

while

5.43.

(i)

(iv)

and

=

Theorem 4

for

T^

of rank

The

*

<1?i

form nos. By

p

=

(A

.

Xt )|

^

KA

,

x)i

=

Appendix

I,

1,2,

in the canonical

p

x t* s

are

Lemma

^

and

We

present the

form (Lemma

botl1 thc

(

H^

5,5) r

)

5.3,

are associate with each other

|

of

of Definition 5.10.

be a fixed operator of finite rank.

x c^P.

(<,)

"^

1

(Lemma

5.40)

we have,

and therefore,

f||f

A

III

Thus, Definition 5.10 gives.

III

X

KA

III

^_(A)

,

r^,x.)[

^ r^i (A xt .

p ?l(A) >(X)

^

P

^ (A)

.

)f

.

This concludes

tit*

proof.

21

122

CROSSED UNITARY SPACES

V.

LEMMA Proof.

5.45.

ot

By assumption by

p*X

For any crossnorm

Lemma

THEOREM

^*X

OU

"jft

,

By Lemma

.

we have

06-^ ?L

5.43 (v),

Thus,

.

5.44.

For a "limited* crossnorm ok

5.16.

the following relation-

,

ships hold: (i)

have

(

Fv^Jft,

f

is a

(ii)

Fv

Proof.

By Lemmas

M

^ i^ot^pTl

fv

.

fCJ&

=

proper subspace

5.43

(iii)

(ii)

follows

^J )*

=

from (

COROLLARY and

5.3.

^j)

(i)

is a

Proof. This follows from

Theorem

The following illustration

is not

=

A^ *J^

)

^*?v

of

consequence is a

^&.rft ^

(Theorem

fv

5.15).

we

p

and

Theorem

proper subspace

pCJ?v 5.13, of

This concludes the proof.

For a limited crossnorm 06

the corresponding sequence

(

F

^/^v

and 5.45, for a certain natural

the fact that

ftq'fi

fv

the spaces

are non- reflexive.

fC/K'

lim ^p=

(

This proves that the spaces

are topoiogically equivalent. Thus,

while

of

.

By Lemma

"X,

,

5.43,

By Theorem

'Xv

at

is

of limited 7* *R

y

.

I.

For ^ we construct

^>J^>

crossnorms. Put

Furthermore,

5.16, for every natural

associate space for the cross- space Fv the associate space

Appendix

without interest:

,....

J^^

2 of

p

linn

(

^X-)'

=

the conjugate and

coincide, while

i
p =

a proper subspace of the conjugate space (Corollary 5.2).

CROSSED UNITARY SPACES

V.

Moreover, for every natural while for

t&)

The structure

^

,

OQ

=

p

{

11.

p

of ideals

a P r P^ r subspace of

*s

> fC

123

they coincide (Theorem 5.13).

(

"Algebraic ideals'* of operators on P& have been considered in the

A

literature. ideal

linear set

on

&J

is

termed an algebraic

if

A

(i)

^5

S.

YAX

implies

Concerning these algebraic ideals

whose proof may be found

An of

A

of operators

fj

in [2a, p. 84 1]

.

^J

for any pair of operators

of interest is the following

X

,

theorem

:

algebraic ideal which does not include

operators consists solely

all

completely continuous operators.

The notion

of an "ideal" of

operators on fv (and also from one Banach

space into another) has been presented here in Definition 4.1. jJJ

of

operators

is

an ideal

if

in addition to condition

(i)

A Banach

space

above, the following

condition holds: (ii)

where

II

||A||

YAX l|<

IIJY/II

represents the

||A|f

||/Xf||

norm

of

A

in

Thus, whenever an ideal

Clearly, an ideal is also an algebraic ideal.

does not include

An

all

operators, all

algebraic ideal however,

may

its

.

yjy

elements must be completely continuous.

not be an ideal as follows from the following

argument:

Let

pjj-

denote the two dimensional Euclidean space and 06 a non-

unitarily invariant (non-uniform)

crossnorm on

fi5

x

f\,^

.

Such crossnorms

Y

,

CROSSED UNITARY SPACES

V.

will be constructed in the last section of this Chapter.

Clearly,

(

ft ^U^v.*. )*

that is, the linear space of all operators on f\^is an "algebraic ideal". it is

o(/

5.3,

(

& ^t^v

and only

is ai* ideal if

)*

if

is unitarily invariant.

DEFINITION

It

A

5.11.

06 -norm

finite

follows that a unitarily invariant

there exists an operator which

if,

that for

|i^OO

&

is significant,

is

crossnorm 06

^5

-norm are

must be also

will be

termed

completely continuous.

is not of finite

the operators of finite

Hence, whenever Ok

crossnorm 06

unitarily invariant

every operator of

if

"significant**

only

Theorem

not an ideal, since by

However

and

is significant if

OC -norm.

It is

06 -norm.

also of finite

significant.

clear

In particular,

for instance, since the identity operator does not belong to the Schmidt-class, that is, is not of finite

-norm, every unitarily invariant crossnorm

<

is significant.

Clearly, the greatest crossnorm

every operator

is of finite

In the

^ -norm (Lemma

^J

is

ot

^

not significant, since

3.5).

present section we shall discuss the structure of the conjugate

spaces of cross-spaces generated by significant unitarily invariant crossnorms. In particular,

we

LEMMA A

operator

Moreover,

Proof.

forms

ai>

shall discuss the relationship to their associate spaces.

Let OC denote a unitarily invariant crossnorm.

5.46.

is of finite

II

A 11^

=

II

ot--norm abs(A) Jl^

By Theorem

ideal.

5.3, the

By Lemma

5.1,

if

and only

if,

abs(A)

is

An

such.

.

space

A

=

of all

operators of

Wabs(A)

and

finite

abs(A)

oC =

-norm

CROSSED UNITARY SPACES

V.

We

125

have, = ||

A |l^

and =

/fA/k Therefore,

II

A /l^

THEOREM

^

Wabs(A) Jk

||

=

abs(A) fl^

/[

.

=

HabsfA)!!^

lllwlll

IJa

This concludes the proof.

A com-

Let oO be a unitarily invariant crossnorm.

5.17.

pletely continuous operator

A

is of finite

canonical representation

A

=

S-c

0^-norm *s

^i^^^c

if

and only

if, its

either finite, or

if

infinite, then

Furthermore,

case

in the last

A

Proof. Suppose that the completely continuous operator finite

We remark

by

of

Ot-norm. Put,

A

finite

is

=

21

B

that

c

=

5.46.

and

<
21 c ^c*

1

oC-norm, since

Lemma

ac

abs(A)

In particular,

^^ =

n

we

-

ZA^

A

= ||

o(/

is unitarily

=

is of finite

is also of

1

.

and therefore

j|

A

= |j

||

B

^,-norm and

A 1^

have,

=

By assumption

K**^ ^

&t

with

abs(B)

ll^-A/f^ Thus, for any natural

^

A^ =

A

'

invariant, hence dU is such by

Theorem

5.2

and

CROSSED UNITARY SPACES

V.

126

Lemma

5.37.

n

function of

Consequently,

(Lemma

Jm^ To prove of finite

rank

5.16); by

^im^

the converse inequality, is

t(XA)

UA^ ||.

a non-decreasing

is

above inequality:

=

a'(Aj

=

&!(A^)

(1

Ajl^ ^

we remark

||

A

fl^

.

that for any operator

X

defined and

*t(XA)

t(XAj This follows from

=

Urn a.w

-

and

i>

|

-

t(XAj m(X)

|

=

t(X

(A^- A)

4

|

m(X) sup a

If)

we have,

n

Clearly, for every

A

-

A^

HI

=

t(XA)|

lim ot!(Aj ^ 4V^O

ot(X)

.

Therefore, also in the limit

^

jt(XA)|

lim

,'(A^) ot(X)

<Mf-^^0

Since the last inequality holds for

all

X

operators

.

of finite rank,

Lemma

5,38

gives, II

A

Urn ct'(AJ ^ ^v-tp

||

*"

-

This proves that for a completely continuous operator 51$, at OP.*"^ finite

o(/-norm

(i)

and

continuous operator

(ii)

A

part of above proof furnishes finite

Conversely, suppose that for a completely

hold.

=

2Lv a C*fw^K; ||

A

||.

4,

relation

lim QC'(A^)

cC" norm and by the first part also i

f

(ii)

holds.

.

(i)

holds.

Hence,

A

The last is of

This concludes the proof.

CROSSED UNITARY SPACES

V.

THEOREM

For a

5.17a.

(Definition 5.11),

(

F^Q^ft,

significant unitarily invariant

THEOREM

5.18.

We assume

that

with

Urn

(i)

K/

& denote

=

Urn a^

,

O

those or

if

also holds.

denotes any fixed

)

Let

.

(ii)

--

a "significant"

crossnorm. Suppose that for every sequence

unitarily invariant

(a^l

^j.

(

all

is either finite,

case

in the last

Furthermore,

throughout the discussion -- cnos in

numbers

crossnorm

represents precisely the space of

)

completely continuous operators whose canonical form infinite then it satisfies (i).

12?

of real

the condition

^(Z

implies

Then,

(

f&

&

CB^

)*

=

^

fv

Together with Ot

Proof.

crossnorm. Hence the value

of

normalized orthogonal sets

(

Let

A

sequence

a^

implies that

A

=

,

(ii)

51 Jm

a

Lemma

a

5.5.

It

is

oC^JC^a-^CAfc cf.

(P.

and

)

(

fU.

i,

a

lim a^

Thus,

satisfies

.....

holds.

associate oC

its

oU-norm and 2l

be of finite

representation of

,

fv

(i)

above.

also a unitarily invariant

does not depend on the

)

.

)

^ftCK> =

.

its

canonical

By Theorem

5.17, the

This by our present assumption

follows that the sequence of operators

w^Pv^Cfw

n

=

is

1|2,....

and this determines a unique operator

A

,

for

fundamental in

which

F\,

Q^ fC

,

128

CROSSED UNITARY SPACES

V.

Consequently,

On

-

JUS

^lim

=

Ajl

.

the other hand,

lim

n^O* Thus,

HI

A

-

A

-

(

=

A^lll

=0

Alfl

Therefore,

III

or

fv^Fv

A

C

)

lim =

A

=

sup a. 4 TWV

(

*v-*0*

.

)

.

K/fl^K

.

On

the other hand since the

^

converse inclusion always holds, we have

(

fvfi^ Pv

""%

= )

PCtS^jPv

This

.

concludes the proof.

12.

We

A crossnorm

whose associate

not a crossnorm.

^

conclude this Chapter with a proof that

least crossnorm.

In fact,

Euclidean space,

n =

fy^Ofv^

is

*

ai*d

we

These by Theorem

we

shall prove that

(but not

2,3,....

1!),

is

not necessarily the

f\^denotes an n-dimensional

if

there is no least crossnorm on

shall actually construct

crossnorms which are not

2.1 furnish

crossnorms (even on

examples

of

^A

finite-

dimensional spaces) whose associates are not crossnorms. By Theorem

none of these crossnorms Clearly,

Let

I

fv

5,4,

hence uniform.

represents the linear set of

all

operators on

R^.

stand for the identity operator.

DEFINITION fv^is

Fy

is unitarily invariant,

.

5.12.

A

non-negative function

termed a quasi-norm

A quasi-norm

is

if it

CC(X)

satisfies conditions

a quasi-cross,norm

if it

(ii)

of

and

operators (iii)

X

on

of Definition 5.7.

satisfies also (iv) of Definition 5.7.

LEMMA

5.47.

V.

CROSSED UNITARY SPACES

Let

X

1

denote an operator of rank

on the two-

1

dimensional Euclidean space pt^. Then, for any complex number

have

-

y(X Proof.

a suitable choice of the coordinate vectors

By

matrix

that the

X

of

/O

may

we can make

we

,

X

replace

complex number with

6X

^

|0| =

,

in pj,^

,

has the form

d)

by

real and

c

tP

c /

I

Since we

a

^

al)

we can assure

29

Since we

.

1

may

replace

we may assume

),

number with

a complex

d

(

tp

d

that

S (P

by

is real

=

|0|

and

d

(

1

)

,

a

^

.

Thus, d

X = c

Let

,

and

We

have:

(X

-

aI)*(X

-a

d c- a

^

al)

,

while

a

is

complex.

denote the trace and the determinant of

|x|

-

( \

are real and

t(X)

spectively. t(

d

,

=

-al

c/

\0

where

X

)

f

=

c"^

)

dx +

+

2 laf" -

Zc^a

X

re-

.

1"

=

|(X-aI)*(X-aI)| Let

a

,

a

,

f

abs(X

-

al)

.

(real and

Then, by

^

a^

and also

,

aj"

,

)

,

Lemma

Y (X Since

J^a-c)/

.

represent the characteristic values of

5.40, -

al)

=

a

+ f

a^

are the characteristic values of

= /

a(a - c)/

,

that is,

a (

a^

=

.

(X

-

/a(a

al)^(X

-

c)|

-

al)

we have,

130

CROSSED UNITARY SPACES

V.

Therefore, -

=

al)

a

+

=

a^

;

+

For

=

a

this

,

a*

+

-4(a^ +

2|a|*

2

2a, a

+

)

|a(a-c)l

becomes

Thus, we wish to establish the following relation:

V

d1

c*- +

'

-

-

2c7?,a

>

cflfta

+

2 la!*"

2 |a(a - c)|

+

that is,

]a(a - c)l

This however,

is

-

)al

v .

obvious, since

?

fa(a-c)|

|el

I

^

V

a)

-

X

denote an operator of rank

|

aj*

c

TCa

-

|a|

This 'concludes the proof.

LEMMA

Let

5.48.

n-dimensional Euclidean space fi^.

we have

-

(X

n

>2

.

complex number

for every

=

n

Since

2

X

has been proven in

is of

rank

1

,

it is

Lemma

of the

X

may

is

completely reduced by

fv'

,

that is,

X

,

X

the operator

identically zero.

^

(X

-

al)

still

has rank

1

,

while in

(pc7 tO

and

.

Fy^j-

tfj

I

fif

fv

the

.

rj'

(and of course

be considered as an operator on fi and as an operator on

In fv

,

We may

5.47.

form

Let fi be a two-dimensional Euclidean space containing Then,

a

>

The case

Proof.

assume

al)

Then

defined on an

1

too)

.

X

is

Therefore, =

^,(X

- al )

+

Y(X

-

al)

^

(X

-

al)

CROSSED UNITARY SPACES

V.

131

and

Thus,

Y*< X

=

t(x)

^ y
^,(X-aI) words

the whole

problem may be considered entirely within

last inequality holds by virtue of

LEMMA such that

g

where

o

O

is a

(X)

-

=

(X)

^ (X) ^

(X)

Lemma

5.48 gives

Thus,

g

"8

=

(X)

The

This concludes the proof.

5.47.

all

FL

^%%,

P x (

^

^

)

^( x

)

O

a

.

.To

=

(l)

It is

prove

obviously sufficient to show that

X

of rank

g

* or a11

for all operators

(X)

Pt^

fi^

complex numbers

it is

by definition of

,

in

quasi-norm, and that

for all operators

(X)

X

for

al)

possesses the cross -property

that

the

(X

taken over the set of

readily seen that that

V

inf A*

is

.

put

=

inf

pf,'

=

(I)

(X)

Lemma

There exists a quasi-crossnorm O defined on

5.49.

We

Proof.

C ^

T

sufficient to prove that

it is

In other

+

>

1

.

stated above.

operators

X

of rank

We

notice first

On

the other hand,

X

of 1

.

rank

1

This completes

proof.

THEOREM QC defined on

Proof.

5.19.

f$J2>

We

&

the

with

,

<

,

ot(l)

=

-g. 1

,

there exists a crossnorm

6

put, Ct(X)

where Q has

Given an

=

(1

-,) g(X)

meaning given

in

Lemma

+ 5.49.

e,fW This concludes the proof.

THEOREM

which

which

^

5.20.

is a

*^

Proof. Since

for

CROSSED UNITARY SPACES

V.

132

=

THEOREM

crossnorm,

By Theorem

7*(I)

ol(l)

is not the least

.

< We

5.21.

>(I)

.

crossnorm on

>

*^(l)

Choose an

.

we construct

5.19

fyJ3

a

for

crossnorm 06

This concludes the proof.

can construct crossnorms on

f(,

fi^ whose

associates are not crossnorms.

^

Proof. Since a

represents the least crossnorm whose associate

crossnorm (Theorem

Theorem

2.1),

our statement

is

also

is

an immediate consequence of

5.20.

COROLLARY

5.4.

We

can construct norms on

/J2)

fy^ which are not

crossnorms and whose associates are crossnorms.

Proof,

we have

fv^is

c&

ot

finite-dimensional.

Thus for any norm OC on

Clearly any crossnorm constructed in

.

Theorem

F^

tf/

5.21

satisfies the required conditions.

REMARK that is

when

fv^,

however, not

I

on

^t/Xw*

difficult to take

belongs to

on fv whose range

only for a finite

(tc)

care of the case

of the above procedure in the course of which

=

oa

This is due to the fact that

represents a Hilbert space Fv

the identity operator is,

n

The arguments presented above exclude the case

5.8.

I

is at least four -dimensional.

n

.

It

n

=

is

replaced by any projection-

oo

by a modification

CROSSED UNITARY SPACES

V.

COROLLARY is

5.5.

None

of the

crossnorms constructed

1

in

Theorem

unitarily invariant (uniform).

Proof. .

By Theorem

5.4, all unitarily invariant

This concludes the proof.

crossnorms must be

33

5.19

APPENDIX Reflexive crossnorms.

1.

Let

we

I

denote a fixed element in a Banach space

f

1JS

F

For

.

^3

put =

<(F)

$

Then,

Thus, we

^*

T^C^* p, P-

may assume

1?*

We

'

By Lemma then

oL

above

^ G>7^>

OC

,

.....

.

A

may

For that

write this

//)

(|f

//

e

7^

therefore be also considered as one in

whenever

o(^ is

a

crossnorm

are also crossnorms on

gate there their mutual relationship.

1#0l

=

fy\\\

space *Q for which

A

^"^ on

9i^ r

1^01^?*'

respectively. Since both cL and OC* are defined on

on

We

.

III

write therefore,

2.11,

!|

,

,

|

180j

bound

its

termed reflexive.

is

/

expression

1

;

The bound represents a norm on 1p

.

in the sense described

An

.

an additive bounded functional on 1>*

is

[l, p. 190]

F(f)

*tf*G>'f&*

,

/

^ ^> (

,

we may

similar statement concerns

06?

,

......

investi-

and 06^

.

the sake of simplicity

1^and T^denote two

we

reflexive

statement that

^and '^are

interpretation

some

of the

shall

assume throughout

Banach spaces. This

reflexive flO, p. 42lJ

.

is

this

Appendix

equivalent to the

For a

slightly modified

statements below also hold for perfectly general

APPENDIX Banach spaces. We also

^

crossnorms

o*

dimensional, we have always

(Theorem

LEMMA

ot(2Z^ S |f^<9 g.

expressionsSl^ljF.

Definition 2.2,

G.)(Z.~9l

o6 '(2E?L,

T

REMARK

1.

f

g^)

The

2.

*$*

At

in

C^-

otfS^^i,

last

Lemma ou"

this point

one

we have

i^*

C

norm

t

?

The answer

is

g.)

C

I

.

.

g^)

7^.^

.

We remark

c

On

GO

OC'^T, Tj

the other hand since by of

such constants, we

This concludes the proof.

06 on

<>

is

^JL

also true for perfectly general

is

tempted

.

of those

Would not the same argument prove 11

t

'P**')^

^

coincides with the

ot

have already shown

which satisfies the inequality:

c

represents the least

8^)

we have

that is,

REMARK Banach space

&

ot" (<-c*\ ^i

Banach spaces,

with

spaces *B,

most general case we have:

constant

is a

)

F.

f^

easy

on

j:,

have,

We

.

g^ be a fixed expression in

f

m(

In the

.

^*L

dL*

1.

Let 2E

Proof. first that

ot^srOO

of the

It is

5.5) that the last relation holds for unitarily invariant (uniform)

crossnorms on R/0Fv

for all

considered are

begin with a few simple statements concerning a norm.

prove (as indicated below) that whenever at least one

is finite

in

135

without having to repeat this assumption each time explicitly.

*X

We to

stipulate that all

I

to

reason as follows: For any

In this inclusion the

norm

for elements

elements when considered

that in general the

"no" since in general

norm

ili

1^

oU coincides

^fr^*^? niay

not

APPENDIX

136

coincide with

where

1^ &d 1@^)**'

(

=

operators on

fj/

while

,

An example

fv 8yr R/

pv/^fv

readily follows

from Theorem

5.15

represents the space of completely continuous

& &^fi,

(

I

is the

)

space of

all

operators on

f(s

.

In fact, in the general case for a given cross-space,its conjugate space contains

The precise conditions

the associate space as a proper subspace.

equality are stated in

Theorem

3.6.

Lemma

4 below proves that whenever the

conjugate and associate space coincide, we actually have

LEMMA

Proof. By

Now

applying

et/"

2.

*

oC/

Lemma

Lemma

1

1,

on

ct ^-06 o
to

^3*0 7^*

.

aJ* ss CX/

.

.

Lemma

Thus,

instead of

for their

OC/

we

get

OU

2.2 implies

ot^

^

W

^ ft

oc'.

Thus, Ot/

O6 *r

X .

&S.

This concludes the proof.

DEFINITION (i)

we have (ii)

(iii)

we have

1.

minimal, OL

^A

if

A crossnorm for

^Ol>v will be termed

06 on

every norm

f&

on

if

06*

1.

For a crossnorm Ct on

are equivalent:

(ii)

(iii)

ot,

Ob O(/

ft*

some crossnorm

/3

on

'?

.

THEOREM

(i)

1

ot.

having an associate property,if for

&

oc

.

reflexive,

ot*

*>O 7^ for which

is

minimal

is

reflexive

has an associate property.

7^01^ the

following statements

APPENDIX

We

Proof.

hand

QtS

Lemma

By Lemma

(i).

By assumption

.

gives oL

1

we assume

we have which

that is,

/,

LEMMA Proof.

^^

.

Let

.

By Lemma

1

O6-

minimal. Thus, oC^

ot^

and

means

is the

3.

is a

2.11,

The last

"5^*0 7?J* is an extension also on

for perfectly general

4.

On

.

associate of oC/

that for

the other

denote a crossnorm on " .> Ot e 06

2,

Thus,

.

A

some crossnorm

ft*

(&

m

Thus,

<*>.

reflexive crossnorm.

^A

*%> *}l

Lemma

of

^

on

.

is

On

the other hand

Lemma

1

gives

also valid for perfectly general Bana'ch

T^*>?^

^O*'?^. An

.

Lemma

By Lemma

application of

2.12,

^^>

2. 11,

Lemma

1,

^ on

which holds

Whenever

for a cross-space

we have,

T^^T^its

06"

conjugate space

~OC. " '!>.

II

f

/I

is the

We least

on

Banach spaces, proves our contention.

coincides with its associate space,

Proof.

on

^ 1^ for

'7

This follows from the following argument: By

LEMMA

OC

^

Thus,

O
and OO have the same

This concludes the proof.

*(i).

/

3.

REMARK spaces.

(iii)

This

(iii).

Thus,

This proves (i)-^(ii).

.

By Lemmas

Otj actf.

is

?&

Then clearly OC

.

&

ot

06

2,

Cfc

& Q(s

We assume OtoC,"

Finally,

137

shall prove (i)^(ii)-*(iii) -(i).

We assume associate

I

recall first that for a fixed

number

'c

f

in a

Banach space

satisfying the inequality

.,

*H3*

,

APPENDIX

138

for all

|F(f)|< cIlFfl

norm

is

be unaffected

if

Since a

that

we

!*L^?^, take 8c)

for all2Li',F-8

G*

<3

^"(^T^jfJ^

2.

we

=

(flFfl

UlF/l/)

l^^,^*

is

the place of

f

only to a dense set in

*s

dense

in

(

and T

in

^k^)*

in

By

^4^,

"^

i

.

Tfe

if

our previous remark,

represents the least constant

will

c

Hence

.

.

Z^c

gT

we see

satisfying the inequality

c

Definition 2.2 however, that constant is

This concludes the proof.

Reflexive cross-spaces.

Here we ular,

gi.)

F

restrict the

V

Ot(5L&Bi*c,

F1^.

clearly a continuous function, the value of the last

By assumption and

I

shall say a few

words about reflexive cross-spaces.

In partic-

point out the relationship between such cross-spaces and those for

which the conjugate and the associate space are identical.

Suppose the cross-space

^^T^is

be considered closed linear subspaces in a reflexive

Banach space

be reflexive.

is

ot

it.

1

while

o(/,

Clearly,

^. ^L/^^nnay not be

sequences

of real

p

>

numbers

1

/

.

xA

We

denote by

for

which 2L

the

1

/

x i|

1^t

andl^may reflexive.

Theorem

prefer to outline the details of the following one directly,

Let

and *9^may

both, *^ and

,

Both 7^

is not true.

such examples can be easily constructed in the light of

EXAMPLE.

IJb

Since a closed linear subspace of

also reflexive flO, p. 423J

The converse statement

flexive and in addition

reflexive.

[4, p.

2

+

be re-

Although

below,

433J

we

.

Banach space

<

must

of all

and where

APPENDIX

(

SL c

may

I

x *|^)

represents the

be interpreted as =

p> 1

Let

elements

(1, 0, 0,

We

respectively.

n> m

^^

....

ip

=

if

(^j

f

i

over the set of

and only

all

)

1

,

denote the sequence

,....

(0, 0, 1,

...

$t (ft)

=

e

=
)

supj

{D fe

,

1

Z

if,

c

|

=

a^,

in

)

with

^

xf

=

l

,

m

4

=

i

and

1

1

4

n

=

y,;|

1

>(r^a $.(fc )4 i

is clearly,

.

a,^^...., a vw

,

we

.

if

Similarly, if

and onlv

$W

j^

we have,

where

I

IfB"!

Putting

.

we have

....

lf>r if -

yA

and

1

3Lw

with the ,

'

get

x,^, + x^^-f

1

- 1,Z .....

i ,j

;

such that B$ll

su PlZ^attytfl

)

1

2^

sequences of real numbers i x^ \

V

= Sc,

The right side

,

and

j

succession for

in

XSZ^.^CH

equality.

Thus,

.

q

J

^[^

1

non- reflexive.

is

1

(0, 1, 0, ....

,

$61

all

the other hand, for ^>

Thus,

1

well-known that

q

and

3

<J?

> (Zj^&li

On

1

p

if

=

$i,and

^_>

1

)

y

=

=

-!-

p

For any constant real numbers

.

taken over

is

+

It is

.

}

have:

Definition 2 .4,

sup

,

=

$t (
By

q

Then,

.

Proof. Let $,

Let

where

1

xc

of

fl,p. 68j.

l

p

of

norm

139

I

=

sup ,

l

|ycl

for

extended

which

Holder's inequality gives: sup

(J^aj)0^|^|^(^^|yj^^

max

I

a.l

.

This concludes the proof of our

.

UO

APPENDIX

I

a <J

Thus, the sequence of expressions 2EL^ m|

a

fundamental

is

4^d^Cp

norm

this is the case, its

if

2L*t *!&&& *fc

P

<

3

i

i

and only

i

if,

>

a^ J

.

Whenever

is clearly

|aj

sup

.

Thus, the well-known non-reflexive space of all

converging towards

a subspace of * lO, p. 423^

p

fl

ot

~ot

1

The

.

[1, pp.

66-67J

numbers may be considered

must not be

last therefore

that the

Banach spaces

reflexive by 7

1%

,

TJJ

are reflexive

,

Then, the associate space forms a"fundamental subspace of

.

We mean

ffaki^* and some

V hereby, that whenever

^ 6^*9^

in

f^

.

w

t*t

F(f^)

=

=

.

F(f

obviously the following:

is

)

be a fundamental sequence of expressions in

and similar ly3C JM

f^ 8^

F

for all

**

**

*

Ta

then

Proof. The meaning of the symbol

(2L

1k/"$for F

^^7^^ for T

a fundamental sequence in

the sequence of corresponding inner products is

and

181J

q

We assume

5.

the conjugate space. in

of real

sequences

P

l

This concludes the proof.

.

LEMMA and

#_>

1

c

Then,

.

.

|

always convergent, to the same limit, independent on the fundamental sequen-

ces of expressions representing

By

assumption,

and

f^

and

Definition 2.2 and continuity of a

satisfying the inequality

^

F

=

F(f) .

is

?()

always

I

^ .

c

f

norm,

dj (F)

Thus,

This concludes the proof,

.

We

denote this limit by

H

dU

(*

for all

ot!{)

=

is the !east

)

F .

in

Hence,

F(f^

number

iiTjf

O,(i)

.

By =

)

c

APPENDIX

LEMMA

Whenever

6.

is reflexive, its

141

I

for a reflexive

crossnorm CU

not in the associate space.

is

"absolutely closed*

1

,

Fe

F

for all flexive.

and

by

9

Lemma

construction, the associate space is

By

l,

p. 5?J

in the associate space.

=

for all

F

f

y"(F

)

=

1

corresponding to

and

^ (F)

This obviously contradicts

^y

,

is

=

re-

F

with

The last implies

in the associate space. *%/

5.

hence

By assumption, our cross-space

contains an element

it

it,

there exists an additive bounded

on the conjugate space such that

Thus,

F(f)

By

of the conjugate

that is, closed in any metric set containing

also in the conjugate space. functional

Q^^^

conjugate and associate space coincide.

Proof. Suppose to the contrary that an element

space

the space

f

(f)

=

1

=

+*

=

F^(f)

1

.

This concludes the

proof.

THEOREM

2.

For a reflexive crossnorm o&

the following statements

are equivalent: 1 ,.

(i)

^^iTP

(ii)

"$<%t'T2

Proof. [lO, pp. 421

is reflexive. is reflexive.

In our proof

and 423]

(a)

A Banach

(b)

A

we

shall use the following propositions stated in

:

space 1^

is reflexive if

and only

if,

if is

closed linear manifold in a reflexive Banach space

reflexive. is also reflexive.

H2

APPENDIX

We with

(iii')

shall prove that

implies

We assume

By

and

and apply

,

while

in conjunction

(iii)

,

they

I

(

(iii),

&

also reflexive.

also reflexive by (b).

must be

oC by

6,

its

Thus, (i)-^(i

,

Tfy

$ V^V

'?*.'>'

to (take the

at!

conjugate

2, Gt'

by IJv and

Thus,

(iii')

both sides of

of)

to the left side of the resulting equality.

(ii)

Hence,

By Lemma

reflexive by (b).

Lemma

=

T?f +,~$)*

we apply

is

obviously be considered as closed

1^ may

Thus, replacing in

obtain

To prove

holds. (iii*),

1?w e

(iii*),

^fl^T^)*

(

^T ^i*?^,

always reflexive. ,by

1>

(ii).

linear subspace s in

(a),

T^^i^^rmist be

closed linear subspace

is

-T (ii)~*(iii) and

(i).

(i).

We assume

(i)

I

Thus,

(ii)

>(iii)

(iii').

we assume

Finally, of (iii).

that

(iii)

and

(iii*)

We

hold.

The resulting equality together with

"^

apply

furnishes

(iii*)

(i).

to both sides

This concludes

the proof.

COROLLARY

2.

For a Hilbert space pv

.

both

and

Uti^ fv

ft<8T Pt

are

non- reflexive.

Proof. =

^V *j* fv (iii') of

By Lemma fv C8L/ fv

Theorem

Theorem

2

2 holds.

if

"^ is reflexive.

is a

proper subspace

does not hold.

Furthermore, by Corollary of

(

^ 8L ^

Consequently, neither

(i)

Thus,

)

or

5.2,

(ii)

of

This concludes the proof.

COROLLARY reflexive

3,

and only

3.

if,

For

a reflexive

crossnorm

every operator from

"T^

into

<X*

,

the space

*9^"of

finite

oL-norm can

APPENDIX norm by operators

be approximated in that

from

1^ into

operators of

Proof. It

^

of finite

finite rank,

The proof

is a

consequence

Theorems

of

3.6 and 2.

does not appear to be a simple matter to state some "reasonable*

such that

its

crossnorm

for which the resulting

cross-space

conjugate space coincides with its associate space.

which follows may be

is

The theorem

of interest although the type of condition stated

herein

from being reasonable.

THEOREM

3.

For a uniformly convex

*2bJ

crossnorm

conjugate space coincides with the associate space

Proof.

Let OC be uniformly convex and

also uniformly convex on

since

rank and every operator

oC/-norm can be approximated in that norm by

of finite

sufficient conditions for a

is far

I

it is

known

that a

Thus, by Theorem

^S^*^.- The

Banach space with

2, its

last

if

06 on 1^

and only

&,"&&, By

the

ok is reflexive.

continuity,

oC

is

cross-space must be reflexive

a uniformly convex

conjugate space must coincide with

That the converse holds for any crossnorm

if,

Tr^.

is the

content of

norm

its

is reflexive.

associate space.

Lemma

4.

This

concludes the proof.

3.

"Limited* crossnorms.

We

conclude this Appendix with a few remarks about a class of reflexive

crossnorms which we on

fv

G> f\,

shall

term "limited*. The notion

of such a

ctpssnorm

has already been mentioned before (Definition 5.10) since in that

1

^

APPENDIX

place

we have found

spaces

it

fv&^fi and

DEFINITION crossnornn on as follows:

convenient to discuss the topological equivalence of the

F^S^fv Let

2.

1

T^

I

Ij^

.

otp (JE^,^

TJi,

We

r a ^y limited

f

crossnornn

ot

.

and 'V^denote any two Banach spaces and oL a define a sequence of functions f cL

g^)

is the least

constant

c

on

J

satisfying the

\

inequality

ZTF.

for all expressions

G-

of

rank 4$ p

Thus, every crossnorm ot on 7JJ507^?

(i)

4.

oCpare crossnorms.

All

(ii)

oC,

(iii)

Ot,

$

^ .....

o6

>

ss

OC

(iv)

.

p

All oC-are reflexive.

(v)

Proof, it is

(ii)

and

(iii)

are a consequence of Definition

also clear that the associate

">

^ oCp ^

is

a

norm

o6

(v).

f

or

of the definitions of

From

Qd/ is

the definition of

This proves

(i).

Definition

^p-

(iv) is

Clearly,

for an expressionZT

It

.

f

t

that

o-

an immediate

remains g^

2.i

Thus,

The verification

ot and Ot- for a given Qt

Otp

From

2.

not smaller than each

the cross-property. O6p possesses

presents no difficulty.

consequence prove

generates a corresponding

V^P3 on

sequence of functions

THEOREM

in

we

to

have,

APPENDIX

I

Thus,

^

for all expressions of rank

p

.

Now,

for a

given^E?^^

g^ Definition

2.2,

gives,

... By

(1), the

.^*

of

f

8j<

t

Lemma

p

(etp)*^^,

5.

W

oO

Now have,

.

This, by Definition Z implies ot

-Thus,

gj

t

p

p

^(oc p )". An

(oCp

OO.

(1) of

^

p

Thus,

coincides with

)

Theorem

4

^(2:^,

*;? gj

<

(v),

is the least

for all expressions of

oC^

proves that (op)'^

Again by Theorem 4

.

(Ot,)

=

<^

p (2^,

f

/i

^

ot p ^ot' ar*d hence

(ii),

p

Definition 2, there

for

which .

therefore oC

(21^,

F.

Gj)

<

.,

/i'(^m,

/'(*^,

^.

By

;

^

for all

p

and an expressionS.i fj& g^

g t)

c

G; of rank

Definition 2. 2, the right side is

Ot-

for all expressions of rank

CX^

suppose that for a crossnorm

exist an expression S.^ ^F.

By

application

.

expressions of rank

^

f

^

rank

Let 06 be a reflexive crossnorm. Then, odp

Proof. Formula

)'

of

G.

crossnorm whose associate

^

g t)

concludes the proof.

1

THEOREM

rank

(*(;:,.

right side is

for all expressionsZlj'^j F.

^Z"?!,

G.)

$* <j

^

')

must

.

we

1

APPENDIX

k6

p e ^c

Thus,

I

for all expressions of rank

^

implies

p

^3

^Qt-.

This concludes the proof.

REMARK

Theorem

zation of

We

on

fvO

crossnorms generated by PC

LEMMA Let

2.1.

conclude this section by pointing out some interesting properties of

the "limited"" Cf

not difficult to see that the last theorem is a generali-

It is

4.

X

(Definition

7.

Let

at*

5.9

"6

where the

.

Then,

last

m

^

p

whose range TJ^(Y)

P

(X

By Lemma

is restricted to the set of all those

sup

Proof. Let

,

is

included in

be the projection of Y)

=

(PX

,

=

Y)

(X

,

=

P*Y)

^ UP

/{I

=

OC(Y)

OC(Y) <X (X) p

consider the sequence of limited crossnorms

the unique self-associate

LEMMA

8.

crossnorm

For every natural

<>

p

Lemma

(X

(3y

+

6*

Ft ,

on

fx>

,

PY)

5.9 (vi), .

Thus,

.

furnish the proof.

i^p/ corresponding

.

,

Y

.

v onT/C'. By

last two relationships and the definition of

We

*Vf&

operators

5.35, the unitary invariance of O(^ implies its uniformity.

OC(PY)

The

5.39).

be a unitarily invariant crossnorm on

P

rank

Lemma

and

crossnorm

be a fixed operator on Fv of finite rank, whose range spans a linear

manifold

of

the unique self-associate

.

to

APPENDIX Let

Proof.

(&

,....,

,

(|^ >(

f

we

I

be an orthonormal set.

,

Using

Lemma

7

readily verify

This concludes the proof.

COROLLARY

p

THEOREM

^

Thus, for any crossnorm p

^

Ols

^.

p

we have &

d

^

'

G*'

implies

C>

C^

^

we have

6^,. Thus,

Gl

G*

(<3l) r

r

6^ for

p

for all expressions of rank

rank

,

Let oC be a crossnorm. Then,

Proof.

G*

But

p

1,2, .....

=

p

1

For every natural

6.

expressions of rank

Now,

for

We have G^ ^ 6^^ $. GT

Proof.

all

ss G"

6*

4.

(

^

p

)'

(

^OC'Odand '

tf

=

G*

Hence also

.

rf(X)

.

C5J,

)=

(x

X)

,

in particular

By Theorem = G* for all

^

<x,(X)

<

3*~

5,

tf

(

G*p ( ^L)* =

/

6*

expressions of

This concludes the proof.

.

REMARK

5.

From Theorem

6

and 4(v)

it

is

clear that

we have constructed

three different reflexive crossnorms which are equal to each other for sions of rank

$

p

,

namely,

6J,

(

,

<>1)

^*

The crossnorms

&

all

and

(

expresc*

are associate with each other. Since, they are also equal for all expressions of

rank

^

we rnay term them "semi-self-associate".

P

REMARK is not

6.

From Remark

determined by the values

(where

p

is

it

5

also follows that a

assumes

crossnorm on

for all expressions of rank

any natural number smaller than the dimension of Fv

)

^,

p

Y

APPENDIX

1U8

1.

A

II

self-associate crossnorm.

THEOREM. Banach spaces

crossnorm on

It is

Tj&f

possible to define a construction which for any two

determines a

TJ^i (without any special restrictions!)

,

^0 1^

this construction

Moreover,

.

when applied

to

unitary spaces, furnishes the usual unique self-associate crossnorm (Definition 5.1

We

andl^emma

5.39)

two 0*

.

precede our proof with the following two Lemmas:

LEMMA

1.

For positive numbers

_

_ a

-I-

,

b

,

_. 4. 4a ^

< ^

b

a

we have,

_

4b

Proof. The proof is immediate.

LEMMA

2.

For any two norms <&*/>>, we have + 2-

Proof. Let in

T^c

^

F

0^^we

fl>

y

<*<'

ft'

By Lemma

*^* be fixed.

1p\

+

/ 1,

for any

non-zero

have,

J_( a \ Thus, Definition 2.2 furnishes the proof.

We

are ready for the proof of our Theorem.

usual pattern, we use a

more

suggestive approach with the introduction of

the notion of a "general crossnorm**.

We

do not attempt here to give a precise

formulation of this notion. For our purpose greatest crossnorm

^

Instead of following the

is sucll

since

it is

it is

sufficient to

remark

uniquely defined on

that the

^ 1^ for

APPENDIX any two Banach spaces T>

crossnorms. When

,

ot is a

II

Similarly,

TJ^.

*h

as well as

general crossnorm,

should also stand for a

oC

general crossnorm with the following significance on

& on 1*0 7fine to

.

>,OT?V

This determines

ot'

on

^f*fo

**

C

T?T

>t

Thus,

'

y

,

^'

,

are general

^"2,

??,O9, The

T^**

3

We

consider

latter

we con-

:

Y+* V X>Y * "T" VTTV .....

.

,

.

are general crossnorms.

We

proceed with our construction:

Put

&

ot

ByLemma2

and

-

>

(0

/

o^-.

+

01^

cow

y

Thus,

*<*,

,,

lim oL^ -

Put,

ot

and <

lim (ot^)'

Since,

and in general,

*~-CoO' we

have,

ot

/i

S

-^ or

Since,

bl^Ot^and consequently

Since

^

-Ot hence, f

Thus,

=0t

.

ot ot'

f or

all

or all

*Vw

'Vv

,

,

we have

we

have,

t'

ot

^^

^

<X/

APPENDIX

150

The obtained general crossnorm It is

with

II

oC- is

defined for any two Banach spaces.

clear that for the case of a unitary space O(

on

By Lemma

K/0 5.39,

Pv

,

& the

resulting

oi*

coincides

hence, is self-associate in the sense of Definition 5.9.

oc coincides with

<*

,

151

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