OPERATORS ACTING ON CERTAIN BANACH SPACES OF ANALYTIC FUNCTIONS

Internat. J. Math. & Math. Sci.

VOL. 18 NO. (L995) I07-0

I07

OPERATORS ACTING ON CERTAIN BANACH SPACES OF ANALYTIC FUNCTIONS

K. SEDDIGHI, K. HEDAYATIYAN^andB. YOUSEFI Departmentof Mathematics,^ShirazUniversity

(Received 3anuary 14, 1993 and +/-n rev+/-sed form September 21, 1993) ABSTRACT. Let be reflexiveBanach space of functions analytic planedomain 12 such that foreveryAini2thefunctionalofevaluation atAisbounded Assumefurther that

X

containstheconstantsand

Mz

multiplication by the independent variable

.,

is bounded operatoron

’.

Wegivesufficientconditionsfor

Mz

^tobereflexive. Inparticular,weprove that the operators

Mz

^on EP{i} andcertain

H{}

^{reflexive. Wealsoprovethatthealgebra}

of multiplication operatorsonBergmanspacesisreflexive, giving simplerproofof result of Eschmeier.

KEY

WORDSAND PHRASES. Banachspaces ofanalytic functions,Smirnovdomain, bounded point evaluation. 1991 SUBJECT CLASSIFICATION. Primary47B37; Sec- ondary47A25.

1 INTRODUCTION.

Letf beabounded domain in thecomplex plane C.

Suppose Z

^isareflexive Banach space consisting of functions thatareanalyticonfl such that

X,

for each

A

in the functional

e(A)" Z

C ofevaluation at

A

givenby

e(A)(f)-< f, e(A)

>-

f(A)

is

bounded,

and if f

X

^thenzf

X. Note

thatthe last condition allowsusto define

Mz ^X X

^by

Mzf ^zf,

^f

X.

It is easy toseethat

Mz

^isactually abounded operatoron

X.

If

Z

isaHilbert space, theoperator

Mz

and many ofitsproperties havebeenstudied in Shieldsand Wallen

[1];

Bourdon andShapiro

[2]. ^We

^{would like}

to givesomesufficient conditionssothat the operator

Mz

becomes reflexive.

Letf beabounded open set inCand let p bearealnumber with

_<

p< ^oo. We denoteby

/.2(1)

^the

^LP-space

^{of the}2-dimensionalLebesguemeasurerestrictedto The space of analytic functionsonfl is denotedby

H(fl)

^andasusual

^g (fl)

^{is theBa-}

nach space of all bounded functions analyticonflequippedwiththe supremumnorm.

Each function

f H(i2)induces

^{aboundedoperator}

Mf’ia( ia(),

g fg, where

L(i2)

^{is the subspace}

of/)’(l)

^consisting of all analytic functions. This space is called the

Bergman

space.

In

this articleweshall prove that the algebra S

(Mf[ f H(fl)}

is reflexive.

We

giveashorterproofofaresult ofJ. Eschmeier

[3]

^{incasefl isa}planedomain.

2 PRELIMINARIES.

In

this section we make a few definitions and set our notation straight. If G is a bounded domain intheplane, the Carathodory hull

(C-hull)

^{ofG isthe}complement of the closure ofthe unboundedcomponentof thecomplementof theclosure ofG. Itcan be described asthe interior of the setofall pointsz0intheplane suchthat

[p(z0)] _<

sup{[p(z)[:

z

G}

^{for all} polynomials p.

An

open set G iscalled a Carathodory domain if it isequalto thecomponent of the Carathodory hullofG thatcontains it.

Forthe algebra

(.’)

^ofall bounded operators on aBanach space

X,

the weak operatortopology

(WOT)

^{istheonein whichanet}

As

converges to

A

if

Aax ^Ax

weakly,x

X.

108 K. SI2D1}IGHI, K. HEDAY,\IIYAN AND B. YOUSEF1

A complex valued function on for which

Cf "

^{for every}

"

^{is called a}

multgherof

"

^{and thecollection}of all thesemultipliers^ISdenoted by

34(2")

^Because

Mz

^isabounded operator^on

Z,

the adjoInt

M; r- X’.

^satisfies

Mz’e(,) ,e(,).

In generaleach multiplier of

Z

^determinesa multiplication operatorM

e

^{defined by}

Me

^Cf,

^,r

^Also

^{Me’e( (,)e()}

^{It is}well known that each multiplier ^isa

bounded analytic function, Shieldsand Wallen

[1]

^Indeed

I()1 ^_<_ ItMelt

^{for each}

infl Also

Mel "

^c

^H()

^{So isabounded} analytic function

Recall that If ^IS a subalgebra of

23(X)

^containing the Identity operator, then

Lat()

^is by definition the latticeof all nvariant subspaces of and Alg

Lat(’)

^is

the algebraof all operators B in

8(,r)

^{such that}

Lat()c Lat(B)

^{We say that}

is

reflexive

^{if Alg}

Lat()

^{Obvmusly areflexive algebra is}

(WOT)-closed

An operator Ain

(.’)

^{IS said tobe}

reflexive

^fAlg

Lat(A)= W(A),

^where

W(A)is

^the

smallest subalgebra of

8(Z)

that contains A and the identity and is closed in the weak operatortopology.

LetA

Al9 Lat(M,)

^{and let}

^N

^{be aweakstar}closed invariantsubspace of

M;

ⁱⁿ

Z"

Then^+/-34

Lat(M,)and

^hence+/-34

Lat(A).

Therefore,

(+/-34)+/- Lat(A*).

^Since

34isweak star

closed,

34

Lat(A’).

^Nowtheone-dimensional span of

e(,)

^isinvariant

under

M;.

Therefore, ^itisinvariant under A*.Wewrite

A*e(A) (A)e(A), ^e a.

So

<

f A’e(,)

>=

(,)f(,); ,

fl. Using the Hahn-Banachtheoremwe seethat the linearspan of

{e(,)}aen

^{isweak stardense in}

^X*.

^Thus

^e 34(Z)

^{andA M}

e.

3 REFLEXIVITY.

In this section weconsider a Banach space of functions analytic on aCarathodory domain and give sufficient conditions for theoperatorofmultiplicationtobe reflexive.

Acircular domain is also considered.

THEOREM

1. Let^glbeaCarathodory^domaineach point of whichisabounded point evaluation forareflexiveBanach space

Z

of functions analyticon f2 whichcon- tains the constantfunctionsand admits

Mz

^{as a} bounded operator.

Furthermore,

if

IIMr, II ^{< CIIpl[n}

for every polynomial p, then

Mz

^lSreflexive.

PROOF. Let A Alg

Lat(M,).

^ThenA

Me

^{forsomemultiplier}

H().

Let

{Pn}

^beasequence ofpolynomials such that

supllpnlln <_

MforsomeconstantM and

pr,(z) (z),

^z

e

fl. Then

IIMp,,ll <_ CIIpnll

n

<_

CM. Since

X

^isreflexive,the unitball of

.

^isweakly compact.

Therefore,

theunitball of

B(.’)

^is

(WOT)

compact.

Wemayassume by passingtoasubsequenceif necessary, that

Mp, X (TOT)

^for

someoperator

X.

Thus

M,,e(,k) X*e(,k)

^{in theweak star}topology. Onthe other hand

M,,e(,X) p,.,(.h)e() (,k)e(,k) Me()in

^{theweak startopologyfor every}

,X I2.

Therefore, X*ea Mea

^{and thusX*}

M. ^{Hence X-- Me}

^on

^Z,

^which

impliesthat

A W(Mz)

^and

Mz

^isreflexive.

Nowwe usethetechniqueoftheproofof Theorem to giveashortproofofaresult of Eschmeier

[3].

^Welet

^B {M.flf ^e g(f2)},

where f isaboundeddomainand

My

actson

L ^(I2).

THEOREM 2. Thealgebra

B

isreflexive.

PROOF. Clearly BC_ Alg

Lat(B).

^Let

^A

hlg

Lat(B).

Becausetheonedimensional span of

e(A)

is invariant under

M

^{for allf in}

^H(fl),

^{it is}invariant under

A*,

and therefore_NextweA_give

M

_afew^forsome_examples^multiplier_ofBanach spaces

.

^Thus

^B

^issatisfying^areflexivethe^algebra.hypothesis^[] ofTheorem 1.

EXAMPLE

^3. Let fl be an arbitrary simply connected Smirnov domain. Let

<p<^{x). Define}

EP(fl)

^tobe the set of allanalyticfunctions fon flsuch that there existsasequence ofrectifiableJordancurvesC1, C2,...ⁱⁿi’l,tendingtotheboundary in thesensethat

Cr,

eventuallysurrounds each compact subdomain of f2 such that

fc,, If(z)lr’ldzl -<

^{M <c. Foragoodsource on}

EP(f)

^seeDuren

[4,

Chapter

10]. ^Every

function of class

E(12)

^hasanontangential limit almosteverywhereon0f,which does not vanishonasetofpositivemeasureunless

f(z)

^0.

Furthermore, fon If(z)lr’ldzl

^<

cx.

It

is convenient toidentify

E(fl)

with its set ofboundaryfunctions. Thus isaclosedsubspace

of/_ev(0fl)

which containsthe set of all polynomials, and henceits closure.

Hence EP(fl)

^isareflexiveBanach space.

Clearly

Mz

^isbounded and

IIMpll <_ IlPlln

for all polynomials p. Now^weshow that

()PEI\TORS ON BANACt-I SPAC!S 109

each point of f s a bounded pointevaluation for

EP(f)

^{For fixed n}

,

choose C >⁰such that

dlst(z,0n)

_>_C Let

_ EP()

^Then

fP

^c

E(f)

^andIthasaCauchy representatmn

Therefore

f(z) ^< (1/2rC) fl

Thus each point off/s abounded pointevaluation for

Ev().

^{Finally,byTheorem31,}

^M,

^sreflexive

Furtherexamplesof Banach spaces satisfying the hypothesis of Theorem will be presented Wealso deduce that

Mz

^actingonthese spacesarereflexive. Webeginwith adefimtion.

DEFINITION4. Let < p < candlet

{/(n)

beasequence ofpomtlvenumbers with

8(0)

^{1. We}consider the space of sequences

f {j;(n)}

such that

ltg I1()[Z()]

^<

.

We shalluse the formal notation

I(z) ,o ](n)z"

^for e

D

the umt disc in C

(See

^Shields

[5]

^for

p=2.).

Let

UP(fl) {I I(z) ,o ](n)z"" llIllp

<

}

^and

Ug() {I

e

HP(Z)l f(z) ,%0 ](n) ^zn

^isconvergent in

D}.

REMARK

5. Define the a-finite measurep onthe positive integersby

p(K)

neg (n) ,K g N.

Because

HP() LP(p)

^we conclude that

HP(fl)

^{is indeed a}

reflexive Banh space.

REMARK

6. If

{fl(n + 1)/fl(n)}

^{isbounded,theoperatorof}multiplication by isaboundedoperatoron

HP().

^Indeed

Mall-

^sup,

Inthefollowing exampleslet q be the conjugate of

p(l’/p(n/- +l/q =1).

EXAMPLE

7. Let

{1/(n)} ^e

^{gq. If f}

^e gP(fl)

^and

^{A e} D,

^we have

Therefore, f is analytic and

]]f

D

N 1{} If

^{p We} conclude that

H() H()

^{c H}

.

Furthermore,each point

oD’

isaboundedpointevaluationfor

HP()

and

so

convergence in

HP(fl)

implies uniform convergenceonD.

EXAMPLE

8.

In

Example 7 assume

(n)

^{for alln.}

^In

^{this case,it follows}

from(l)

^that

Ilf

^g

c _fl

^{for anycompact}

^{K cD,}

^{whereC dependson K.}

c-+)_

EXAMPLE

^9. Letp> 1. Also suppose that sup,

(n) =l+l/n).

^{Itcaneily beseenthat}

a(Mz).

^Since

^M,

^isacontrition,

D

is aspectral setfor

M,

and

]IM[ ]lp]D

for every polynomial p.

By

Theorem 1,

M,

actingon

H()is

^reflexive.

The domains considered inTheorem wereCarathodorydomains. Wenowextend the conclusion of thisTheoremtoaccular

domNn,

thatis,anydomNnobtainedby removingafinite number of disjoint subdiscs from the open unit disc. In Seddighiand Yousefi

[6]

^{wehaveprovedthe}analogueof thefollowingtheorem foraHilbertsubspe of

H().

^Fortheproofcombinethetechniquesof theproofofTheorem withSeddighi andYousefi

[6,

^Theorem

5.1].

THEOREM

10. Let beacirculdomNn each point of which is abounded point evaluationforareflexiveBh subspace

X

of

H()

^which

^contNns

the constants and

ts

multiplicationbytheindependentvariablez,

M,,

^abounded operator.

Furthermore,

suppose that

]Mp N lp In

for every polynomial p. Then

M,

isreflexive.

WepresentanexampleofaBanach spacesatisfyingthehypothesisofTheorem10.

EXAMPLE

11.

Let

be acircular domain and < p <

.

^Since

L()

^is

closed in

(), ()

is reflexive.

By

Lemma3.7ofGarnett

[7]

every point of is abounded point evaluation for

(). ^It

^isalso clear that

l]Mp] N lip n

^{for every}

polynomial p.

By

Theorem 4 themultiplicationoperator

Mz

^on

L()

is reflexive.

ACKNOWLEDGEMENT.

Researchofthe first authorwaspartiallysupported by agrant

(no. 67-SC-520-276)

from ShirazUniversityResearch Council.

ll0 K. SEDDIGHI, K. HEDAYAT[YAN AND B. YOUSEFI

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