Dual Operator Algebras[;] Sufficient Conditions for the Classes $bold A sb {n,aleph sb 0}$

DUAL OPERATOR ALGEBRAS; SUFFICIENT CONDITIONS FOR THE CLASSES

YOUNG SOO JO^I), IL BaNG JUNG2 ,3) AND HUNG HWAN

LEE

Let H be a separable, infnite dimensional, complex Hilbert space and let £(H) denote the algebra of all bounded linear operators on H.

A dual algebra is a subalgebra of £(H) that contains the identity opera- tor I'H and is closed in the ultraweak operator topology on £(H). This notion of dual algebras was introduced by S. Brown in [6J, where he proved that every subnormal operator has a nontrivial invariant sub- space. The theory of dual algebras is deeply related to the classes

which will be defined below and the study of the problem of solving systems of simultaneous equations in the predual of a singly generated dual algebra (see [1], [3J and [4]). This theory is applied to the study of invariant subspaces and dilation theory. In particular, in [7] Chevreau- Exner- Pearcy obtained some characterizations of operators in the class

AI,No'

In addition, Exner-Jung [10J defined certain Hereditary prop- erties concerning a minimal isometric dilation of T in A and obtained some characterizations for membership of the class

As a sequal study of them we define a certain index

of a contrac- tion T and discuss relationship between the index

and the classes

An,No

in this paper.

The notation and terminology employed here agree with those in [2J, [4J and [18J. We recall nonetheless them for the convenience of the reader.

Suppose that A is a dual algebra in C(H). Let Cl = C1(H) be the ideal of trace class operators in £(H) under the trace norm and let

Received October 8, 1994.

1991 AMS Subject Classification: 47D27. 47A20.

Key words: Dual algebra, dilation. defect indices.

1)

partially supported by University affiliated research Institute, Korea Research Foundation, 1993.

2)

partially supported by KOSEF, 94-0701-02-01-3.

3)

partially supported by the Basic Science Research Institute Program, Ministry

of Education, 1993.

.LA denote the preannihilator of A in Cl. Let QA denote the quotient space Cl /.LA. One knows that A is the dual space of QA and that the duality is given by

(1) < T, [L] > = trace(TL), TEA, [L} E QA.

Furthermore, the weak* topology that accrues to A by virtue of this duality coincides with the ultraweak operator topology on A (cf. [8]).

For T E £(1£), let AT denote the dual algebra generated by T. For vectors x and y in 1£, we write, as usual, x ® y for the rank one operator in Cl defined by

(x ® y)(u) = (11., y)x, 11. E 1£.

We shall denote by D the open unit disc in the complex plane C and we WFite l' for the-boundary of D. For 1

p

we denote the usual Lebesgue function space by LP ~ LP(T).. For 1 ~ P ~

we denote by HP = HP(T) the subspace of LP consisting of those functions whose negative Fourier coefficients vanish. One knows.that the preannihilator .L(H=) of H= in Ll is the subspace HJ consisting of those functions 9 in HI for which analytk extension 9 to JIlsatisfies 9(0) = o. It is well known that H= is the dual space of Ll / HJ.

Let us recall that any contraction T can be written as a direct sum T = T

EEl Tz, ^where T

is a completely nonunitary contraction and Tz is a unitary operator (cf. [18]). If Tz is absolutely continuous or acts on the space (0), T will be called an absolutely continuous contraction.

The following provides a good relationship between the function space H= and a singly generated dual algebra AT.

FOIA~-NAGY

Let T be an absolutely con- tinuous contraction in £(1£). Then there is an algebra homomorpmsm

«PT : H= --+ AT defined by «PT(f) = I(T) such that (a) «PT(l) = In, «PT(e) = T,

(b) II«PT(f)1I ~ 11/11=, ^{lE Hoo,}

(c) «PT is continuous if both Hoo and AT are given their weak*

topologies,

(d) the range of «PT is weak* dense in AT,

(e) there exists a bounded, linear, one-to-one map <PT : QT --+

Ll / HJ ^{such that} <PT = «PT, and

(f)

is an isometry, then

is a weak* homeomorphism of Hoo onto AT and ,pT is an isometry of QT onto Ll / HJ.

Suppose that m and n are any cardinal numbers such that

n

~No.

A dual algebra A will be said to have property (Am,n) if every m

system of simultaneous equations of the form

where {[L

is an arbitrary m x n array from Q,A, has a solution

O<i<n

{Xi}o<i<m, {Yj-}o<j<n consisting of a pair of sequences of vectors from 11. For brevity, we-shall denote (An,n) by (An). The class A(11) consists of all those absolutely continuous contractions Tin £(11) for which the functional calculus

: Hoo ---4 AT is an isometry. Furthermore, we denote by A

,n(11) the set of all T in A(11) such that the algebra AT has property (Am,n)' We write simply Am,n for Am,nCH.) unless we mention otherwise.

If M is a semi-invariant subspace for T E £(1i) (i.e., there exist invariant subspaces .NI and .N2 ^for T with.N

:J .N2 ^{such that} M =

.NI e _.N2), T M denotes the compression of T to M. In other words, TM = PMTIM, where PM is the orthogonal projection whose range is M.

Throughout this paper, we write N for the set of natural numbers.

For a Hilbert space K, and any operators Ti E £(K,), i = 1,2, we write T

T

if T

is unitarily equivalent to T

Recall that T E C. o ^{if IIT*n x} lI ---40 for any x E 11. We say T E Co.

if T* E C. o. And we denote that Coo = Co. n e. o.

Let T be a contraction operator in £(11) and we denote by BT E

£(K,+) a minimal isometric dilation of T, where (2)

It follows from Wold decomposition theorem that

(3) BT = STffiRT,

where ST E £ (UT ) is the unilateral shift part and RT E £(RT) is the

residual part. Furthermore, it follows from (3) that B T ⁼ Sf EEl R T ^is

a minimal coisometric extension of T*. Let UT E £(K) be a minimal unitary dilation of T, where

(4)

00

K = V ^U!}'H.

n=-oo

Let £T = (UT - T)'H and let £r- = (UT - T*)'H. Recall that dim£T

= d

and dim£r- = dr-, which are called defect indices of T and T*, respectively.

Throughout this paper we denote a index by

(5) OtT = dim(£T n 'RT).

PROPOSITION 1. Suppose that T is a contraction on 'H. Let M be an invariant subspace ofT apd let f be a restriction ofT to M. Then OtT $ OtT·

Proof. Let BT E £(K+) be a minimal isometric dilation of T with BT = ST(f)RT, where ST E £ (UT) is the shift part and !!-T E £('RT) is the residual part. Then B

is the isometric dilation of T. Hence there

exi~s

K+. E Lat(BT) such that BTIK+. is a minimal isometric dilation of T with the decomposition

BTIK~ = _ST(f)RT E UT (f) 'RT'

We write BT = ^BTIK+.. Since every minimal isometric dilations (or minimal unitary dilations) of a given contraction are unitary equivalent each other, it is sufficient to show that £T C £T and 'Rfo C RT. Since one is obvious, it is sufficient to show that 'R _T C 'RT.

Let x E 'RT and let x = ^{s(f)r E} UT (f) 'RT. Since BT = ^{BTIK+., we}

have

Bfn= ( ' : ~)

relative to a decomposition K+. (f) (K e ^{K+.), where An} is some bounded operator from

to K e

for any n E N. Futhermore, we have

(6) ^{IIxll2 $ IIxll2} + IIAnxll2 =

Rj;nx

+ IIAnxll2

= IIBirx (f) A nxll 2

= IIBfnxll2 $ IIx1l 2 .

Hence Anx ⁼ 0 for any n E N. This proves that

(7)

IIsll2+ IIrll2 = IIxll2 = IIRfnxll2 =

Bjrx

2

= IIBrxll2 + IIAnxll 2 = IIBfnx EEl Anxll 2

=

Bfnx

2 ₌ IISfnsll2 + IIRTnrll2

= IISfnsll2 + IIrll2.

Letting n

on the right side of (7), we have that s = o. ^{So x E RT.}

Hence the proof is complete. 0 The following is fundamental.

PROPOSITION 2. Suppose that T

is a contraction on 11i with

=

- n

ri, i = 1,· .. , n. Let T = EElf=l Ti. Then at = Ei=l ri.

If T E C. o, then

= ^O. If T is an isometry, then

= ^{O. In}

this paper we study completely nonunitary contraction operators with

aT

=F O. For example, if S is a unilateral shift operator of multiplicity mEN, then as. = m.

LEMMA 3. Suppose that T is a completely nonunitary contraction on 11 with aT- = n E N. Suppose that {ul:h<l:<n is an orthonormal set in L.T- n RT-. Let us denote - -

(8)

and (9)

Then (10)

M (V ^{{u I:}) =} V R~ (V ^{uI:})

1:=1 n=-oo 1:=1

n=-oo

M(V

^{{UI:}) = LEElk=l} M (UI:).

1:=1

Proof. Let

and v be any two vectors in {uA:}19~n. Since

L.T- = (UT- - T*)11,

(11)

there exist sequences {h n} C 1£ and {gn} C 1£ such that u = lim(UT* - T*)h n

n

and

which implies that

(R}.u, v) = (R}.(lim(UT* - T*)hn),lim(UT* - T*)gn)

n n

= limlim(U~(Ur- - T*)hm, (Ur- - T*)gn)

m n

= limlim((U~*hm,gn) - (U;;-IT*h m,gn)

m n

- (U;:tlh m, T*gn) + (U~T*hm, T*gn»

= liriiIim((T*khm,gn)- (T*"-I T *hm,gn)

- (T*"+Ih m,T*gn) + (T*"T*h m,T*gn» ⁼ 0, k = ±1, ±2,··· . Note that (u,v) = ^O. Hence we have

J.. M(uj), i i=-

j and the proof is complete. 0

We write

for the usual multiplication function on L2('f).

LEMMA 4. Suppose that T is a completely nonunitazy contraction on 1£ with ar- = n E N. Then

RT*

Meit

o ...

(n) relative to some decomposition

L 2('f)

L 2('f)

, _...

(n)

Proof. The idea of this proof comes from [18, p.83-89]. Let 9 =

.er- n Rr-. By Lemma 3, there exists a set of non-zero vectors {u It:} k=1 in 9 such that

(12)

Now we consider a fixed Uk from the above. Let {Ed

be the spectral family associated with RT-IM(Uk)' Note that Etis-absolutely continuous function of t. For

E IC, rn,

E N, we have

(13)

where

27r .

(RToa, R'T- a) = 1 ei(m-n)td(Eta, a)

t

= 10 ei(m-n)t p(t) dt,

p(t) = dt(Eta,a). d

There exists a unitary q> from M(Uk) onto L2(Ok), where Ok = {t: t E (0, 271"),p(t) > O}.

Since Uk is a nonzero *-cyclic vector for RT-IM(Uk), we have Ok (0,271"), k = 1,'" ,no Since we can assume that L

(1') is identified with L2(~h),

Hence we have

(14) RT-I (I:EBk=lM(Uk)) ~!Jei' EB"'EBMeit,

v

(n)

and the proof is complete. 0

The following lemma comes from [15] (or [17]).

LEMMA 5. Let T be in A with RT the unitary piece of its minimal coisometric extension and let RT has multiplicity at least n on 1'. Then T E An,No'

The above lemmas prove the following theorem.

THEOREM 6. liT E A with aT- = n, then T E An,No'

Note that there are several characterizations of unitary operators in

the classes An,No in [11], [12] and [13]. Hence the following corollary

provides several sufficient conditions for operators in the classes An,No'

COROLLARY 7. Let T be a contraction operator on H. Suppose that

T ~ (~1 ;2)

relative to some decomposition, where T

is a completely nonunitary in A with

= m and T

is a unitary in An. Then T E _Am+n,~o.

Proof. Since T

.is unitary, T

E

and RT

has multiplicity at least n on T (cf. [13]). Since RT

RTl ^ffi RT2 (cf. [5]), the residual part of T

ffi T

has multiplicity at least m + non T. Hence by Lemma 5 we have T E

0

Let T be a contraction on H. Recall that if dT <

and d

<

00,

then T is a Fredholm operator and the Fredholm index ind(T) is equal to d

d

(cf. [16)). Recall from [9, Theorem 2.4] that if T E An,~o n C.o, then ^d

^d

~ n.

PROPOSITION 8. Let T E C. _o with d

<

Then ind(T*)

Proof. If dT. =

since ind(T*) =

the proposition holds. Hence we can assUIDe that dp <

By [9, Lemma 1.3] we have

(15) R y ^~

where B is the bilateral shift operator of multiplicity one and n indT*. By Lemma 4, there exists an invariant subspace M of R y such that

RylM ~

where r =

Hence

~ r. tl

Recall that a completely nonunitary contraction T E ,C( H) is said to be of class Co if there exists a non-zero function u E HOO(T) such that the functional calculus u(T) = O. Lef

be the unilateral shift operator on a Hilbert space of multiplicity n. In [14, Theorem 1], they proved a generalization of the theorem that if

is unitarily equivalent to an operator matrix form

(S~I) ;)

relative to a decomposition M ffi N, then E is in a certain class Co.

We give a simple proof as following.

COROLLARY

9. Suppose that

is the unilateral shift operator of multiplicity n for a positive integer n and

relative to a decomposition M EEl.N. Then E E Co.

Proof. Since

E C.

E E C.

Let us consider the canonical decomposition

(16)

(s

s(n)

~ 0

we have dE; -dE] = O. Hence it follows from (15) that E

E Co., which implies that the canonical decomposition in (16) must be E = El.

So E E Coo. By [18, Theorem VI.5.2] E E Co. Hence the proof is complete. 0

ACKNOWLEDGEMENTS.

The authors owe a debt of gratitude to Pro- fessor George Exner for the contributions he made to this paper.

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Young Soo Jo

Keimyung University

n ^{Bong Jung &} Hung Hwan Lee

Kyungpook National University