I. Dual Algebras

In thes e lectures we will be interested in bounded linear operators acting on a

separable, comple x Hilber t space . Mostly , w e wil l b e intereste d i n th e cas e i n

which the Hilbert space is infinite dimensional, but in several topics to be treated,

the finite-dimensiona l cas e i s interestin g an d sometime s eve n important . Thus ,

throughout the book X wil l denot e a separabl e comple x Hilber t spac e whos e

dimension i s less than or equal to K 0, an d J?(X) wil l denot e th e algebra of al l

bounded linear operators on X. Furthermore , throughout the book X wil l denote

a separable, complex Hilbert space whose dimension is equal to tf

0

.

With X a s above , le t I K = K(X ) denot e th e norm-close d idea l o f compact

operators in «£?( X\ an d recall that K contains all other proper ideals in &(X ).

The quotien t spac e S£(X )/IK i s a C*-algebr a whic h is trivia l when dim X N

0

and is called the Calkin algebra when X = X. Th e projection of J?(X) ont o the

Calkin algebra is denoted by m. If K e K(X\ w e write the polar decompositio n

K = UP, wher e P = (K*K) 12. ' Then , o f course , P e K(X) an d thu s ha s a

diagonal matrix Diag(X

1?

X2 ,...) relativ e to some orthonormal basis { en}™xl fo r

X, wher e X

n

^ 0 fo r n e N . Thi s correspondenc e P - Diag(A x, A2 ,...) ca n be

used to define th e Schatten /?-ideals as follows.

For p\, defin e V

p

= V p(X) t o b e th e se t o f al l K = UP belongin g t o

K(X) suc h that E^ \

p

n

+ 00.

PROPOSITION

1.1. Forp 1, tf

p

(X) is a (two-sided) ideal in &(X) which is a

Banach *-algebra under the norm \\K\\p = {E^LjX;} 1^. The family {V

p

(X)}p1

is increasing, and if 1 p q, then \\K\\ ||JC|| ^ HJq ^ for all Kin V p(X).

We will mostly be interested in ^ X(X\ th e trace-class, and, to a lesser extent,

#2(X), th e Hilbert-Schmidt class.

DEFINITION

1.2. I f X =£ (0 ) is a finite-dimensional Hilber t space, then the only

nonzero idea l i n £P(X) i s J?(X ) itself . Nevertheless , fo r p ^ 1, w e will writ e

Vp(X) fo r £P(X ) equippe d wit h th e norm | | H ^ defined similarl y t o what was

done above, an d we call ^i(X) an d #

2

(X) th e trace-class and Hilbert-Schmidt

class on Jf , respectively , equipped with the /race norm || ||j an d Hilbert-Schmidt

norm || || 2.

l

http://dx.doi.org/10.1090/cbms/056/01