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 ) .