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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