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
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
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
X2 ,...) relativ e to some orthonormal basis { en}™xl fo r
X, wher e X
^ 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
= 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^ \
+ 00.
1.1. Forp 1, tf
(X) is a (two-sided) ideal in &(X) which is a
Banach *-algebra under the norm \\K\\p = {E^LjX;} 1^. The family {V
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.
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 #
(X) th e trace-class and Hilbert-Schmidt
class on Jf , respectively , equipped with the /race norm || ||j an d Hilbert-Schmidt
norm || || 2.
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