2 DUAL ALGEBRA S
The following proposition show s where the trace-class gets its name.
PROPOSITION
1.3. With X as always, there is a continuous linear functional tr :
#x( X) - * C on the trace-class with the property that if {e m}mGM is any orthonor-
mal basis for X, then tr(K) = L
mG M
(Kem, e m) and
UK)\\\K^V^(K^K)l/1)
for every K in V x(X). Finally, if T e &(X) and K e ^(X), then ti(TK) =
tt(KT).
In thes e lecture s operator s o f ran k on e wil l pla y a distinguishe d role , s o i t i s
worthwhile t o review som e elementary fact s abou t them . I f x, y e X, w e denot e
by x 0 y th e rank-on e operato r define d b y ( x ® y)(u) = (u, y)x fo r ever y u i n
X. On e easily checks that if A e i f (X) , the n
A(x ® y) = (Ax) ®y an d (x ® y)(A) = x $ (^*) -
PROPOSITION 1.4. Fo r a/ / x , j i « Jf , x ® j e V
l9
tr( x ® 7 ) = (x , y), and
\\x®y\\i = \\x®y\\ = \\x\\\\y\\.
We now recall some standard dualit y results (cf. [18, p. 40]).
PROPOSITION
1.5. With X as always, the dual space €X(X)* of the Banach
space ^
X
{X) can be identified with J£(X). This duality is implemented by the
bilinear functional
(T,K) = tr(TK), T^Sf(X), K e «\ .
In particular, we have that
|| 71 =sup{|(r) tf|:*etf
1 )
11*11^1}
and
Mi - sup{|7\*)| : Te#(jT), | | r | | l } .
Throughout th e remainde r o f thes e lecture s w e wil l identif y ^ X(X)* wit h
i f ( J f ) withou t furthe r comment . I n particular , thi s dualit y give s t o £P(X) a
weafc* topology, which i s characterized b y th e fac t tha t a ne t {T x} i n JP(X) i s
weak* convergen t t o an operato r F
0
i f an d onl y if fo r ever y K e V
l9
tr(T xK) - »
trCToA'). We now briefly revie w some other important topologie s on JSf ( X ).
The ultraweak operator topology on J^(Jf ) i s tha t locall y convex , Hausdorf f
topology determine d b y sayin g tha t a ne t {T x} i n if(Jf ) converge s ultraweakl y
to T
0
i f an d onl y if , fo r ever y pai r {x n}™msl and {y n)™-x o f sequence s fro m X
such tha t L^ilKII
2
0 0 an d L ^ l b J
2
00 , w e hav e E^i(F
x
x
n
, g - *
^x(T0xn,yn).
The weak operator topology (WOT) on & (X) i s that locally convex, Hausdorf f
topology determined b y saying that a net { Tx} i n i f (X) WOT-converge s to r
o
i f
and onl y if, fo r al l x, y & X, (T
x
x, y) - * (r
o
x,), or equivalently ,
t r ( r
x
( x ® ^ ) ) - + t r ( r
o
( x ® 0 ) .
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