4

DUAL ALGEBRA S

(e) th e smalles t weak*-close d subalgebr a j /

r

o f if ( Jf) tha t contain s a give n

operator T an d \

x

. Th e dua l algebr a sf

T

i s calle d th e dual algebra generated

byT;

(f) th e algebra of all analytic Toeplitz operators relative to some fixed orthonor -

mal basis for Jf. Thi s is a particular case of (e).

EXAMPLE

1.10. Ther e exists a commutative dual algebra that is not closed in the

WOT. Indeed , le t K b e a n operato r i n ^ ( J f ) tha t i s not o f finit e rank , an d le t

Ji denot e th e kerne l o f th e weak*-continuou s linea r functiona l induce d o n

e(3e) b y K, i.e. ,

Note tha t Ji i s weak*-closed but not WOT-closed. (For, if Ji wer e WOT-closed ,

then K woul d induc e a WOT-continuous functiona l o n Ji, an d thu s by Proposi -

tion 1.7, K woul d necessaril y b e o f finit e rank. ) No w conside r th e algebr a

j / c &(JP& JtT) defined b y

•*-{(£ x )

: X e c r 6

- 4

It is easy to see that s/ i s weak*-closed but not WOT-closed .

PROBLEM

1.11 . Does there exist a singly generated dual algebr a stf

T

tha t i s not

WOT-closed? Westwoo d [33a ] has show n tha t ther e exists a (WOT-closed) singl y

generated dua l algebra on which the weak* topolog y and WOT do not coincide .

DEFINITION

1.12 . I f y i s any subse t of i?(Jf) , the n ± &= {K e e

x

: (K, S)

= 0 , S e S?}

y

i s th e preannihilator o f 6P, which is a (weakly closed ) subspace of

^x. (Her e we mention fo r th e first tim e the weak topology on c €1 tha t accrue s to it

as the predual of ££(X\ W e also note that in these lectures the word "subspace "

always means "norm-closed linea r manifold". )

DEFINITION

1.13 . I f Sf i s a subse t o f ^f(Jf), the n a subspac e Ji o f X i s a

nontrivial invarian t subspac e (n.i.s. ) fo r & i f (0 ) # Ji # J T an d SJi c Ji fo r

every S e^ an d a nontrivial hyperinvariant subspace (n.h.s.) for SP if it is a n.i.s.

for th e dua l algebr a SP\ Th e lattic e o f al l invarian t subspace s fo r Sf wil l b e

denoted b y Lat(^ ) an d th e lattic e o f al l hyperinvarian t subspace s o f Sf b y

Hlat(^).

The followin g elementar y proposition show s th e importance o f preannihilator s

and rank-on e operators.

PROPOSITION

1.14 . A dual algebra s#zg{X) has a n.i.s. if and only if

x

jtf

contains some rank-one operator x ® y, and a n.h.s. if and only if

x

(sf) contains

a rank-one operator.

DEFINITION

1.15 . I f ^ i s a subset o f Se(X) w e write AlgLat(^) fo r th e dua l

algebra (whic h i s WOT-closed ) consistin g o f al l T e i f ( j r ) wit h Lat(T ) D

Lat(^). A (necessaril y WOT-closed ) subalgebr a @ o f J£?(Jf ) i s reflexive i f