PRELIMINARY RESULTS AND BACKGROUND
5
the identity operator on X* (o r to the canonical embedding of X int o X**).
For an y v in X* E X, w e define the "trace" by letting
tr(i) =/,* .
Clearly,
(0.6) | t r ( u ) | H |
A
.
If v = E" x*S x, , we have tr(v) = E " x*(x,). Similarly , if £ : X - A" is a finit e
rank operato r wit h associate d tenso r v i n X * ® X, we define it s trac e simpl y b y
setting tr(v) = tr(v) . This has the usual properties of th e trace. For instance, let u:
X - Y an d w: Y - X b e operators . I f on e o f the m i s o f finit e rank , the n i t i s
easy t o chec k tha t tr(wvv ) = tr(vvw) . One o f th e mai n difficultie s her e i s tha t thi s
notion o f trac e doe s no t mak e sens e in genera l o n N(X
9
X) sinc e th e correspon -
dence between X* § X an d N( X, X) i s (in general) not injective . The problem is
that (0.6 ) does not impl y |t r v\ N(v) fo r all finite rank operator s v: X - X. Th e
next lemma implies, however, that this is true if X possesse s the A.P.
LEMMA 0.4 . Let u: X - Y be an approximable operator. Let v be in Y* 0 X,
and let v: Y - X be the associated finite rank operator. Then
(0.7) \lr(uv)\N(v)\\u\\.
PROOF.
Le t v = E f
TJ *
$ z, . Assume that N(v) 1.
There is an expansion of the for m
oo
(0.8) V * e X, o(x) = £ \ nyn*(x)xn
« = 1
with X
n
e R , j„ * e B Y+, x
n
e tf^, an d suc h that E|AW| 1.
By modifyin g (\ n) i f necessary , w e ca n assum e tha t x
n
- » 0 whe n H - oo .
Hence, we may assume that th e sequence (xn) an d th e set ( zi)i^ ar e included i n
a compact se t A' .
Let a
e
: X - y b e a finite rank operator such tha t
sup|||a
e
x - «x| | | x e A j e.
Then
(0-9) t\Ky:(«x.-«s*)\*-
1
From (0.8), it follows immediately (since a
£
is of finite rank) tha t
(010) tr(a e£) = £ \„*(«
£
xj.
On th e other hand, since { zv..., z
n
} c A' , we have
| tr(ap) - tr ( uv) | - | £ *?*(V , " */ ) I
C e with C = E h f | | .
l
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