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
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