2
PRELIMINARY RESULT S AND BACKGROUN D
In general , w e refe r th e reade r t o th e book s o f Lindenstraus s an d Tzafrir i
[L-Tl, L-T2 ] fo r unexplaine d terminolog y an d fo r mor e backgroun d i n Banac h
space theory .
b. A n introductio n t o tenso r products . Th e approximatio n property . Nuclea r
operators. Let X, 7 b e Banach spaces. Consider the linear tenso r product X ® Y.
Each element v in X ® Y ha s a representatio n
n
(0.1) v = £ x
t
® y
t
fo r x
t
e X, y
t
e 7 .
I
Moreover, w e may clearly regard v also as a bilinear for m o n X * X 7 * mappin g
an element (x*, *) in X* X 7 * intoEx*(;c z)^*(j/).
We ca n als o associat e t o o a finit e ran k operato r v: X* - 7 whic h i s
a( A"*,
Ar)-continuous;
this operator i ; is defined b y
vx* e x*, £(**) = 53 **(*/).yl..
The injective tenso r product nor m | | | |
v
i s defined o n ! 0 7 a s follows:
(0.2) H v = H k ^ )
= sup{X*(x,)*(^)|x *
G
*** ' ^ *
G B
r*)-
This nor m coincide s wit h th e operato r nor m o f u o r wit h th e nor m o f u a s a
bounded bilinea r for m o n I * X 7* .
The projective norm | | | |
A
is defined b y
(0-3) lkl U = inf|i:i|x,||||^||}
where the infimum run s over all possible representations (0.1). Clearly, we have
(0.4) I M I V ^ H I A .
Let a b e a norm o n X ® 7 . We denote by X ®
a
7 th e space X ® 7 equippe d
with th e nor m a . W e cal l a "reasonable " i f th e bilinea r map s (x , y) - x S y
from X X 7 int o X 0 7 an d (*', y') - + JC ' 8 / fro m X * X 7 * int o (X ®
a
7) *
are both o f nor m 1. This implies a(x ® y) = \\x\\ \\y\\. With thi s terminology, it
is eas y t o chec k tha t th e norm s | | | |
A
an d | | | |
v
are , respectively , th e larges t an d
the smallest reasonable norm o n 7 .
The completio n o f A" ® 7 wit h respec t t o ||||
v
(resp . | | || A) i s calle d th e
injective (resp . projective ) tenso r produc t o f X an d 7 an d i s denote d b y X ® 7
(resp. X i 7). I t i s easy t o se e that an y elemen t v i n X 8 7 ca n b e represente d
as a series v = Ef
3
x
n
® yn wit h x
n
i n X , in 7 suc h tha t E||xJ | ||jJ| oo , and
||y||
A
is the infimum o f EJ° ||jtJ| || j„|| over all such representations.
The spac e X i 7 ca n be described easily ; by (0.2) it can be identified wit h th e
closure i n B(X*, 7 ) o f al l th e o(X*, X)-continuou s operator s o f finit e rank .
However, i t i s more delicate to describe X E 7 a s a space of operators . By (0.4),
we have a norm decreasin g map /: X ® Y -* X k 7 . But , in general , this map i s
not injective.
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