characterizations of bergman space toeplitz operators with ... · characterizations of bergman...

J. reine angew. Math. 617 (2008), 1—26

DOI 10.1515/CRELLE.2008.024

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin � New York 2008

Characterizations of Bergman space Toeplitzoperators with harmonic symbols

By Issam Louhichi at Toledo and Anders Olofsson at Stockholm

Abstract. It is well-known that Toeplitz operators on the Hardy space of the unitdisc are characterized by the equality S �

1TS1 ¼ T , where S1 is the Hardy shift operator. Inthis paper we give a generalized equality of this type which characterizes Toeplitz operatorswith harmonic symbols in a class of standard weighted Bergman spaces of the unit disccontaining the Hardy space and the unweighted Bergman space. The operators satisfyingthis equality are also naturally described using a slightly extended form of the Sz.-Nagy-Foias functional calculus for contractions. This leads us to consider Toeplitz operators asintegrals of naturally associated positive operator measures in order to take properties ofbalayage into account.

0. Introduction

Let nf 1 be an integer. We denote by AnðDÞ the Hilbert space of all analytic func-tions f in the unit disc D with finite norm

k f k2An

¼ limr!1

ÐD

j f ðrzÞj2 dmnðzÞ:

The measure dm1 is the normalized Lebesgue arc length measure on the unit circle T andfor nf 2 the measure dmn is the weighted Lebesgue area measure given by

dmnðzÞ ¼ ðn� 1Þð1 � jzj2Þn�2dAðzÞ; z A D;

where dAðzÞ ¼ dx dy=p, z ¼ xþ iy, is the planar Lebesgue area measure normalized so thatthe unit disc D has area 1. The space A1ðDÞ is the standard Hardy space, the space A2ðDÞis the unweighted Bergman space and in general the space AnðDÞ is a so-called standardweighted Bergman space. The norm of AnðDÞ is also naturally described by

k f k2An

¼Pkf0

jakj2mn;k;

Research of second author supported by the M.E.N.R.T. (France) and the G.S. Magnuson’s Fund of the

Royal Swedish Academy of Sciences.

where mn;k ¼ 1=k þ n� 1

k

� �for kf 0, using the power series expansion

f ðzÞ ¼Pkf0

akzk; z A D;ð0:1Þ

of the function f A AnðDÞ (see [11], Section 1.1).

The shift operator Sn on the space AnðDÞ is the operator defined by

ðSn f ÞðzÞ ¼ zf ðzÞ ¼Pkf1

ak�1zk; z A D;ð0:2Þ

for f A AnðDÞ given by (0.1). Recent work has revealed that the so-called dual shiftoperator

S 0n ¼ SnðS �

n SnÞ�1

plays an important role in the study of shift invariant subspaces of AnðDÞ. The operator S 0n

is a weighted shift operator on AnðDÞ which acts as

ðS 0n f ÞðzÞ ¼

Pkf1

mn;k�1

mn;kak�1z

k; z A D;ð0:3Þ

on functions f A AnðDÞ given by (0.1).

In recent work [17] on characteristic operator functions we have used the fact that theBergman shift operator Sn satisfies the operator equality

ðS 0nÞ

�S 0n ¼ ðS �

n SnÞ�1 ¼Pn�1

k¼0

ð�1Þk n

k þ 1

� �Skn S

�kn :ð0:4Þ

For full details of the proof of formula (0.4) we refer to [17], Section 1.

For n ¼ 1 equality (0.4) simply says that the Hardy shift operator S1 is an isometrymeaning that S �

1S1 ¼ I . We notice that for n ¼ 2 equality (0.4) can be written as

ðS 02Þ

�S 0

2 þ S2S�2 ¼ ðS �

2S2Þ�1 þ S2S�2 ¼ 2I :ð0:5Þ

A similar to (0.5) looking operator inequality has appeared in work of Shimorin [18] onapproximation theorems of so-called wandering subspace type and in work of Hedenmalm,Jakobsson and Shimorin [10] on weighted biharmonic Green functions. It is known that theshift operator S : f 7! zf in the class of logarithmically subharmonic weighted Bergmanspaces in the unit disc with weight functions that are reproducing at the origin satisfies theinequality

kSf þ gk2e 2ðk f k2 þ kSgk2Þ

for all functions f and g in the space (see [10], Proposition 6.4), and that this last inequalityis equivalent to the operator inequality

2 Louhichi and Olofsson, Bergman space Toeplitz operators

ðS �SÞ�1 þ SS �e 2I

(see [18], Proof of Theorem 3.6). This last inequality shows a close resemblance to equality(0.5).

We shall study in this paper bounded linear operators T A L�AnðDÞ

�on the Berg-

man space AnðDÞ satisfying the operator equality

ðS 0nÞ

�TS 0

n ¼Pn�1

k¼0

ð�1Þk n

k þ 1

� �Skn TS

�kn in L

�AnðDÞ

�:ð0:6Þ

We shall show that this equality (0.6) characterizes the Toeplitz operators on AnðDÞ withbounded harmonic symbols within the class of all bounded linear operators in L

�AnðDÞ

�(see Theorem 6.1). By a Toeplitz operator on the Bergman space AnðDÞ, nf 2, withbounded harmonic symbol h in D we mean the operator Th in L

�AnðDÞ

�defined by the

formula

ðTh f ÞðzÞ ¼ÐD

1

ð1 � zzÞnhðzÞ f ðzÞ dmnðzÞ; z A D;ð0:7Þ

for functions f A AnðDÞ. For n ¼ 1 we can identify a bounded harmonic function h in D

with its boundary value function in LyðTÞ by means of the Poisson integral formula (see[12], Lemma III.1.2). In this way formula (0.7) gives a standard formula for a Hardy spaceToeplitz operator. We notice that for n ¼ 1 equality (0.6) reduces to S �

1TS1 ¼ T and that inthis way a well-known characterization of Toeplitz operators on the Hardy space A1ðDÞ isrecovered.

The operators T A L�AnðDÞ

�satisfying equality (0.6) turn out to admit also a simple

description using a slightly extended form of the Sz.-Nagy-Foias functional calculus forcontractions on Hilbert space taking into account also powers of the adjoint of the opera-tor. Let us describe briefly this functional calculus. Let T A LðHÞ be a contraction on aHilbert space H meaning that T is an operator on H of norm less than or equal to 1.We shall use the notation

TðkÞ ¼ T k for kf 0;

T�jkj for k < 0;

�ð0:8Þ

which is standard in dilation theory. From the existence of a unitary dilation of T it followsthe existence of a positive LðHÞ-valued operator measure doT on the unit circle T suchthat Ð

T

eiky doTðeiyÞ ¼ TðkÞ; k A Z:

By an approximation argument the operator measure doT is uniquely determined by thisaction justifying the notation doT (see Section 4).

We shall need an appropriate method of summation and as a matter of conveniencewe shall use the Cesaro summability method. Let f A L1ðTÞ be an integrable function onT. The N-th Cesaro mean sN f of f is defined by the formula

3Louhichi and Olofsson, Bergman space Toeplitz operators

ðsNf ÞðeiyÞ ¼ ðKN � f ÞðeiyÞ ¼P

jkjeN

1 � jkjN þ 1

� �ff ðkÞeiky; eiy A T;

where ff ðkÞ ¼ÐT

f ðeiyÞe�iky dy=2p is the k-th Fourier coe‰cient of f and

KNðeiyÞ ¼P

jkjeN

1 � jkjN þ 1

� �eiky; eiy A T;

is the N-th Fejer kernel. It is well-known from harmonic analysis that such means havegood approximation properties coming from the fact that fKNgNf1 is a well-behaved ap-proximate identity (see [12], Section I.2).

Let us return to a contraction operator T A LðHÞ. It is fairly straightforward to seethat if f A CðTÞ is a continuous function on T then

ÐT

f ðeiyÞ doTðeiyÞ ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ff ðkÞTðkÞ in LðHÞð0:9Þ

with convergence in the uniform operator topology in LðHÞ (see Section 4). If the opera-tor measure doT is absolutely continuous with respect to Lebesgue measure on T we canmore generally integrate essentially bounded measurable functions f on T and for suchfunctions f A LyðTÞ the above limit (0.9) holds with convergence in the strong operatortopology in LðHÞ (see Theorem 4.1). It should be mentioned here a result of Sz.-Nagyand Foias related to the structure of the minimal unitary dilation which gives that theoperator measure doT is absolutely continuous with respect to Lebesgue measure on T ifT A LðHÞ is a completely non-unitary contraction (see [19], Theorem II.6.4).

Let us return to a bounded linear operator T A L�AnðDÞ

�on the Bergman space

AnðDÞ. We shall show that such an operator T satisfies equality (0.6) if and only if it hasthe form of a functional calculus integral

T ¼ÐT

f ðeiyÞ doSnðeiyÞ in L

�AnðDÞ

�of a function f A LyðTÞ relative to the Bergman shift operator Sn, and that the normequality kTk ¼ k f ky holds (see Theorem 5.1).

We discuss also these operator integrals arising from the shift operator

S : f 7! zf ; f A H;

on a Hilbert space H of analytic functions on the unit disc D such that the operator S is acontraction on H. For such an operator S the operator measure doS is always absolutelycontinuous with respect to Lebesgue measure on T and we have the norm equality

��ÐT

f ðeiyÞ doSðeiyÞ�� ¼ k f ky; f A LyðTÞ


(see Theorem 5.2). We mention that this last norm equality also follows by a result ofConway and Ptak [9], Theorem 2.2, using that the HyðDÞ-functional calculus for the shiftoperator S is isometric. We also show that the operatorÐ

T

f ðeiyÞ doSðeiyÞ in LðHÞ

is compact if and only if f ¼ 0 (see Theorem 5.3). These two results generalize to the con-text of Hilbert spaces of analytic functions well-known properties of Toeplitz operators onthe Hardy space of the unit disc.

Above we have described two seemingly di¤erent looking characterizations of theclass of operators T A L

�AnðDÞ

�satisfying equality (0.6), namely, first as Toeplitz opera-

tors Th on AnðDÞ with symbols h that are bounded harmonic functions in D, and then asfunctional calculus integrals of functions f A LyðTÞ with respect to the shift operator Sn.The correspondence between these two classes of operators is given by

Th ¼ÐT


�AnðDÞ

�;

where h ¼ P½ f � is the Poisson integral of f A LyðTÞ. We interpret this equality that somesort of balayage is going on and that Toeplitz operators should be considered di¤erently asintegrals of naturally associated positive operator measures in order to make this process ofbalayage more transparent. We discuss some rudiments of such a theory in Section 7 in thispaper.

Let us describe briefly how we analyze the operators satisfying equality (0.6). Thefunction space AnðDÞ is equipped with a natural group of translations te iy parametrizedby elements on the unit circle eiy A T. It is a general fact that a bounded linear operatoron such a space can be decomposed into homogeneous parts with respect to such a groupof translations and the operator can then be reconstructed from its homogeneous parts as alimit in the strong operator topology using an appropriate method of summation analogousto what is usually done in harmonic analysis. We discuss this construction in Section 2.By homogeneity type of arguments an operator satisfying (0.6) can be described as anoperator with all of its homogeneous parts satisfying (0.6), and for an operator of a fixeddegree of homogeneity we can apply shifts or backward shifts in order to reduce to the caseof an operator which is homogeneous of degree 0 which is nothing else but a Fourier mul-tiplier. Arguing this way we show that an operator T A L

�AnðDÞ

�satisfies equality (0.6) if

and only if every k-th homogeneous part Tk of T is a constant multiple of SnðkÞ (see The-orem 3.1). We then use this analysis to arrive at the descriptions indicated above.

Partially driven by classical work of Brown and Halmos [7] on algebraic propertiesof Toeplitz operators on the Hardy space much e¤ort has been put into the study of thecorresponding questions for Toeplitz operators on the unweighted Bergman space A2ðDÞ:Axler and Cuckovic [6] have solved the commutativity problem of when two Toeplitz op-erators with harmonic symbols commute. Ahern and Cuckovic [2] have solved the productproblem of when the product of two Toeplitz operators with harmonic symbols is again aToeplitz operator with harmonic symbol; see also Ahern [1]. From this point of view wegive in this paper the generalization of Theorem 6 of Brown and Halmos [7] to the Berg-man spaces AnðDÞ.

The authors thank Elizabeth Strouse for useful discussions.


1. Preliminaries

We collect in this section some constructions and formulas involving the shift opera-tor Sn on AnðDÞ in order to make the presentation in later sections more e‰cient. We alsosketch some background on positive operator measures.

Shifts and backward shifts on the Bergman space. The adjoint shift operator S �n in

L�AnðDÞ

�acts as

ðS �n f ÞðzÞ ¼

Pkf0

mn;kþ1

mn;kakþ1z

k; z A D;ð1:1Þ

on functions f A AnðDÞ given by (0.1). Notice that the space

kerS �n ¼ AnðDÞmSn

�AnðDÞ

�consists of the constant functions in D. The operator Ln ¼ ðS �

n SnÞ�1S �n in L

�AnðDÞ

�is the

left-inverse of Sn with kernel kerLn ¼ kerS �n . The operator Ln acts as

ðLn f ÞðzÞ ¼f ðzÞ � f ð0Þ

z¼Pkf0

akþ1zk; z A D;

on functions f A AnðDÞ given by (0.1). In other words, the operator Ln is the backwardshift operator on AnðDÞ.

As is apparent from the introduction we shall make use of the dual shift operator

S 0n ¼ SnðS �

n SnÞ�1 in L�AnðDÞ

�which is a weighted shift operator on AnðDÞ whose action is given by (0.3). Notice thatðS 0

nÞ� ¼ ðS �

n SnÞ�1S �n ¼ Ln in L

�AnðDÞ

�and that the operator

L 0n ¼

�ðS 0

nÞ�S 0n

��1ðS 0nÞ

� ¼ S �n in L

�AnðDÞ

�is the left-inverse of S 0

n with kernel kerL 0n ¼ kerS �

n consisting of the constant functions.

Positive operator measures. Let ðX ;SÞ be a measure space consisting of a set X

and a s-algebra S of measurable subsets of X , and let H be a Hilbert space. By apositive LðHÞ-valued operator measure dm we mean a finitely additive set functionm : S ! LðHÞ such that mðSÞ is a positive operator in LðHÞ for every S A S and themarginal set functions mx;y, x; y A H, defined by

mx;yðSÞ ¼ hmðSÞx; yi; S A S;

are all complex measures in the usual sense. Notice that this amounts to saying that dmx;x isa finite positive measure for every x A H. If f is a measurable complex-valued functionsuch that f is integrable with respect to dmx;y for all x; y A H we say that f is integrable

with respect to dm and we define the integralÐX

f dm as an operator in LðHÞ by the require-ment that


�ÐX

f ðsÞ dmðsÞx; y�

¼ÐX

f ðsÞ dmx;yðsÞ; x; y A H:

It is straightforward to see that every bounded measurable function f on X is integrablewith respect to dm and that we have the norm bound

��ÐX

f ðsÞ dmðsÞ��e kmðX Þk k f ky;ð1:2Þ

where

k f ky ¼ inffc > 0 : mðfx A X : j f ðxÞj > cgÞ ¼ 0 in LðHÞg

is the essential supremum of f (see [15], Section 1).

A positive operator measure dm with mðXÞ ¼ I such that mðSÞ is an orthogonal pro-jection in LðHÞ for every S A S is called a spectral measure. We record also that a mea-surable function f is integrable with respect to a spectral measure dm ¼ dE if and only if itis essentially bounded and that equality always holds in (1.2) in this case. Positive operatormeasures dm such that mðXÞ ¼ I are in the literature sometimes called quasi or semi spec-tral measures.

2. Decomposition of an operator into homogeneous parts

We shall discuss in this section a decomposition of an operator into homogeneousparts with respect to translations. This decomposition is of somewhat independent interestand we discuss it here in some more generality than needed for our applications.

Let X be a Banach space and denote by LðXÞ the space of bounded linear operatorson X. We assume that the circle group T operates on elements in X in such a way that themap

t : T C eiy 7! te iy A LðXÞ

is continuous in the strong operator topology in LðXÞ, the operator t1 is the identity oper-ator I in LðXÞ and

te iðy1þy2Þ ¼ te iy1 te iy2 in LðXÞ; eiy1 ; eiy2 A T:

Notice that this group structure simplifies the continuity requirement of the map t to therequirement that

lime iy!1

te iyx ¼ x in X; x A X;

with convergence in the norm of X. Notice also that the operator norms kte iyk areuniformly bounded by the Banach-Steinhaus theorem. Let T A LðXÞ and consider theoperators


Tkx ¼ 1

2p

ÐT

e�ikyte�iyTte iyx dy; x A X;ð2:1Þ

for k A Z. The integral in (2.1) is interpreted as an X-valued integral of a continuous func-tion. Clearly Tk is an operator in LðXÞ and the operator norm of Tk is bounded by aconstant times the norm of T . A change of variables shows that the operator Tk has thehomogeneity property that

te�iyTkte iy ¼ eikyTk in LðXÞ; eiy A T:

We call such an operator homogeneous of degree k with respect to translations. It isstraightforward to see that if Tj A LðXÞ is homogeneous of degree j and Tk A LðXÞ is ho-mogeneous of degree k, then the product TjTk A LðXÞ is homogeneous of degree j þ k.We notice also that if X ¼ H is a Hilbert space and T A LðHÞ is homogeneous of degreek, then the adjoint operator T � A LðHÞ is homogeneous of degree �k.

Let us return to the above general discussion. Keeping in mind that for x A X the map

T C eiy 7! te�iyTte iyx A X;

is a continuous X-valued function on T we have by a standard argument that

Tx ¼ limN!y

1

2p

ÐT

KNðeiyÞte�iyTte iyx dy in X; x A X;

whenever fKNgNf1 is a suitable approximate identity (see [12], Section I.2). In particular,choosing KN as the N-th Fejer kernel we obtain that

Tx ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �Tkx in X; x A X;

meaning that the operator T can be reconstructed from its homogeneous parts Tk by meansof Cesaro summation in the strong operator topology in LðXÞ.

We shall apply the above discussion when X is the Bergman space AnðDÞ and thetranslations te iy are acting as

ðte iyf ÞðzÞ ¼ f ðe�iyzÞ; z A D;

on functions f A AnðDÞ as is customary in harmonic analysis. Notice that here the trans-lations te iy are unitary operators in L

�AnðDÞ

�and that this gives that

t�e iy ¼ te�iy in L�AnðDÞ

�; eiy A T:

It is straightforward to see that an operator T A L�AnðDÞ

�is homogeneous of degree 0 if

and only if it acts like a Fourier multiplier in the sense that

ðTf ÞðzÞ ¼Pkf0

tkakzk; z A D;


for f A AnðDÞ given by (0.1). Boundedness of the operator T corresponds to the multipliersequence ftkgkf0 being bounded. The Bergman shift operator Sn is homogeneous of degree1, and such homogeneity has also the dual shift operator S 0

n. The left-inverses Ln and L 0n are

homogeneous of degree �1.

We can think of an operator T in L�AnðDÞ

�as given by an infinite matrix ftjkgj;kf0

relative to the orthogonal basis fekgkf0 of monomials

ekðzÞ ¼ zk; z A D;

for kf 0. The action of the operator T is then described by

TðekÞ ¼Pjf0

tjkej in AnðDÞ; hTek; ejiAn¼ tjkmn; j:

In this context the m-th homogeneous part Tm of T corresponds to the matrixfdm; j�ktjkgj;kf0 obtained from the matrix ftjkgj;kf0 by putting all elements outside thediagonal m ¼ j � k equal to 0.

3. Powers of shifts and adjoint shifts

Recall the notation (0.8). We shall prove in this section that an operatorT A L

�AnðDÞ

�satisfies equality (0.6) if and only if every k-th homogeneous part Tk of T

is a constant multiple of SnðkÞ (see Theorem 3.1). The proof will proceed in several steps.

We first show that powers of shifts and adjoint shifts satisfy equality (0.6).

Proposition 3.1. The operators SnðkÞ in L�AnðDÞ

�for k A Z all satisfy equality (0.6).

Proof. We first show that Skn satisfies (0.6) for kf 0. We have that

ðS 0nÞ

�Skn S

0n ¼ ðS �

n SnÞ�1S �n S

kn SnðS �

n SnÞ�1 ¼ Skn ðS �

n SnÞ�1:

By equality (0.4) we conclude that

ðS 0nÞ

�Skn S

0n ¼ Sk

n

Pn�1

j¼0

ð�1Þ j n

j þ 1

� �S jnS

�jn

!¼Pn�1

j¼0

ð�1Þ j n

k þ 1

� �S jnS

kn S

�jn :

This shows that Skn satisfies (0.6).

By a passage to adjoints we see that also the operators S�kn for kf 0 satisfy equality

(0.6). This completes the proof of the proposition. r

We next consider a Fourier multiplier in L�AnðDÞ

�satisfying equality (0.6).

Lemma 3.1. Let T A L�AnðDÞ

�be homogeneous of degree 0, and assume that T sat-

isfies equality (0.6). Then the operator T is a constant multiple of the identity operator I .


Proof. As we have pointed out in Section 2 the assumption of homogeneity of de-gree 0 means that the operator T A L

�AnðDÞ

�acts as

ðTf ÞðzÞ ¼Pjf0

tjajzj; z A D;

on functions f A AnðDÞ given by (0.1) for some bounded sequence ftjgjf0 of complexnumbers.

Let us prove that tj ¼ t0 for jf 0. The assumption that T satisfies equality (0.6)means that

hTS 0n f ;S

0n f iAn

¼Pn�1

k¼0

ð�1Þk n

k þ 1

� �hTS�k

n f ;S�kn f iAn

; f A AnðDÞ:ð3:1Þ

Let f A AnðDÞ be a function of the form (0.1). By (0.3) we have that

hTS 0n f ;S

0n f iAn

¼Pjf1

tjm2n; j�1

mn; jjaj�1j2 ¼

Pjf0

tjþ1

m2n; j

mn; jþ1

jajj2:ð3:2Þ

By (1.1) we have that

ðS�kn f ÞðzÞ ¼

Pjf0

mn; jþk

mn; jajþkz

j; z A D;

for kf 0. Computing scalar products and summing we now have that

Pn�1

k¼0

ð�1Þk n

k þ 1

� �hTS�k

n f ;S�kn f iAn

ð3:3Þ

¼Pn�1

k¼0

Pjf0

ð�1Þk n

k þ 1

� �tjm2n; jþk

mn; jjajþkj2

¼Pjf0

Pminð j;n�1Þ

k¼0

ð�1Þk n

k þ 1

� �tj�k

mn; j�k

!jajj2m2

n; j;

where the last equality follows by a change of order of summation. Varying the functionf A AnðDÞ of the form (0.1) the above equalities (3.1), (3.2) and (3.3) give that

tjþ1

mn; jþ1

¼Pminð j;n�1Þ

k¼0

ð�1Þk n

k þ 1

� �tj�k

mn; j�k

ð3:4Þ

for jf 0. We introduce now the function

f ðzÞ ¼Pjf0

tj

mn; jz j; z A D:


Notice that the sum in (3.4) is equal to the j-th coe‰cient in the power series expansion ofthe function

Pn�1

k¼0

ð�1Þk n

k þ 1

� �zk

!f ðzÞ; z A D:

Multiplying (3.4) by z j and summing for jf 0 we have that

f ðzÞ � t0

z¼ Pn�1

k¼0

ð�1Þk n

k þ 1

� �zk

!f ðzÞ ¼ 1 � ð1 � zÞn

zf ðzÞ; z A D:

We can now solve this last equation for f ðzÞ to conclude that

f ðzÞ ¼ t01

ð1 � zÞn ¼ t0Pjf0

1

mn; jz j; z A D:

This shows that tj ¼ t0 for all jf 0. This completes the proof of the lemma. r

We shall need also the following lemma concerned with operators T A L�AnðDÞ

�of

positive degree of homogeneity.

Lemma 3.2. Let T A L�AnðDÞ

�be homogeneous of degree mf 0. Then the range of

T is contained in the range of Smn .

Proof. Since the operator T is homogeneous of degree m it must equal its m-th ho-mogeneous part Tm by considerations from Section 2. Recall also the matrix representationof the m-th homogeneous part from Section 2 saying that the operator T ¼ Tm acts as

Tek ¼ tkþm;kekþm; kf 0;

relative to the orthogonal basis of monomials ekðzÞ ¼ zk. It is easy to see that the operatorSmn has closed range. We conclude that the range of T is contained in the range of Sm

n . Thiscompletes the proof of the lemma. r

We can now describe a general operator T A L�AnðDÞ

�satisfying equality (0.6).

Theorem 3.1. Let T A L�AnðDÞ

�. Then the operator T satisfies equality (0.6) if and

only if every k-th homogeneous part Tk of T is a constant multiple of SnðkÞ.

Proof. The if-part is evident by Proposition 3.1 and summability considerationsfrom Section 2.

Let now T A L�AnðDÞ

�be a general operator satisfying (0.6). We first show that ev-

ery k-th homogeneous part Tk of T satisfies (0.6). Recall that the operators Sn and S 0n are

both homogeneous of degree 1. Using (0.6) and these homogeneity properties we computethat


ðS 0nÞ

�te�iyTte iyS0n ¼ te�iyðS 0

nÞ�TS 0

nte iy ¼ te�iy

Pn�1

k¼0

ð�1Þk n

k þ 1

� �Skn TS

�kn

!te iy

¼Pn�1

k¼0

ð�1Þk n

k þ 1

� �Skn te�iyTte iyS

�kn in L

�AnðDÞ

�:

We can now pass to the k-th homogeneous part using (2.1) to conclude that Tk satisfies(0.6) for every k A Z.

We shall next show that if an operator Tk A L�AnðDÞ

�is homogeneous of degree

kf 0 and satisfies (0.6), then Tk is a constant multiple of Skn . Recall that ðS 0

nÞ� ¼ Ln. We

first show that LknTk satisfies (0.6). Since Tk satisfies (0.6) we have that

ðS 0nÞ

�LknTkS

0n ¼ Lk

n ðS 0nÞ

�TkS

0n ¼ Lk

n

Pn�1

j¼0

ð�1Þ j n

j þ 1

� �S jnTkS

�jn

!:ð3:5Þ

By Lemma 3.2 the range of Tk is contained in the range of Skn . This gives that the operator

Tk factorizes as Tk ¼ Skn L

knTk. By (3.5) we conclude that Lk

nTk satisfies (0.6). The operatorLknTk is also homogeneous of degree 0, and Lemma 3.1 applies to give that Lk

nTk ¼ cI . Weconclude that Tk ¼ Sk

n LknTk ¼ cSk

n .

We now finish the proof of the theorem. We know that each homogeneous part Tk ofT satisfies (0.6). By the result of the previous paragraph we have that Tk is a constant mul-tiple of Sk

n if kf 0. Assume next that k < 0. Then T �k is homogeneous of degree jkj and by

a passage to adjoints we see that also T �k satisfies (0.6). The result of the previous paragraph

applies to give that T �k is a constant multiple of Sjkj

n . Taking adjoints again we see that Tk isa constant multiple of S�jkj

n ¼ SnðkÞ. This completes the proof of the theorem. r

4. Functional calculus for contractions

In this section we shall discuss an extension of the Sz.-Nagy-Foias functional calculusfor contraction operators on Hilbert space taking into account also powers of the adjoint ofthe operator. Let us proceed to describe this functional calculus.

Let T A LðHÞ be a contraction operator on a Hilbert space H meaning thatkTke 1. A classical result of Sz.-Nagy concerns the existence of a unitary dilation of T ,that is, a unitary operator U A LðKÞ on a larger Hilbert space K containing H as aclosed subspace such that

T k ¼ PHUkjH in LðHÞ

for kf 0, where PH denotes the orthogonal projection of K onto H (see [19], Chapter I).A related construction is that of a positive LðHÞ-valued operator measure doT on the unitcircle T having Fourier coe‰cients

ooTðkÞ ¼ÐT

e�iky doTðeiyÞ ¼ T�k for kf 0;

T jkj for k < 0:

�


Notice that doT is uniquely determined by its Fourier coe‰cients. This operator measuredoT can be obtained as the compression to H of the spectral measure for a unitary dilationU A LðKÞ of T . Alternatively, the operator measure doT can be constructed usingoperator-valued Poisson integrals (see [15] for a construction in the context of the n-torusTn).

As indicated in Section 1 integration with respect to doT is a continuous linear map

CðTÞ C f 7!ÐT

f ðeiyÞ doTðeiyÞ A LðHÞ

of CðTÞ into LðHÞ of norm equal to 1. Let f A CðTÞ and consider the Cesaro meanssN f ¼ KN � f of f defined as in the introduction. It is well-known that the Cesaro meanssNf converges to f in CðTÞ (see [12], Section I.2). Integrating with respect to doT we arriveat the limit assertion that

ÐT


PjkjeN

1 � jkjN þ 1

� �ff ðkÞTðkÞ in LðHÞ

with convergence in the uniform operator topology in LðHÞ, where ff ðkÞ denotes the k-thFourier coe‰cient of f .

If the operator measure doT is absolutely continuous with respect to Lebesgue mea-sure on T we can more generally integrate functions in LyðTÞ. We shall next study thecorresponding approximation property of such integrals. Here should be mentioned a resultof Sz.-Nagy and Foias which asserts that the spectral measure of the minimal unitary dila-tion of a completely non-unitary contraction is absolutely continuous with respect to Leb-esgue measure on T (see [19], Theorem II.6.4). This result gives by a compression argumentthat the operator measure doT is absolutely continuous with respect to Lebesgue measureon T if T A LðHÞ is a completely non-unitary contraction.

We shall need the following lemma from integration theory.

Lemma 4.1. Let dm be a positive LðHÞ-valued operator measure on a measure space

X such that mðX Þ ¼ I , and let f be a measurable function on X which is square integrable

with respect to dm. Then��ÐX

f ðsÞ dmðsÞx��

2

eÐX

j f ðsÞj2 dmx;xðsÞ; x A H:ð4:1Þ

If dm ¼ dE is a spectral measure, then equality holds in (4.1).

Sketch of proof. We consider first the case when dm ¼ dE is a spectral measure. Forf a simple function the assertion of equality in (4.1) can be verified by straightforwardcomputation. The case of a general function f A LyðdEÞ then follows by approximationby simple functions.

Let now dm be a positive operator measure such that mðXÞ ¼ I in LðHÞ. By a resultof Neumark [14] the operator measure dm can be dilated to a spectral measure dE (see for-mula (7.1) in Section 7). Using this dilation we can verify inequality (4.1) for f bounded.The case of a general function f follows by an approximation argument. r


We can now prove the corresponding approximation property for integrals of LyðTÞ-functions.

Theorem 4.1. Let T A LðHÞ be a contraction operator such that the operator mea-

sure doT is absolutely continuous with respect to Lebesgue measure on T, and let f A LyðTÞ.Then

ÐT


PjkjeN

1 � jkjN þ 1

� �ff ðkÞTðkÞ in LðHÞ

with convergence in the strong operator topology in LðHÞ.

Proof. Let KN be the N-th Fejer kernel and denote by sN f ¼ KN � f the N-th Ce-saro mean of f A LyðTÞ. It is well-known that lim

N!ysN f ¼ f pointwise a.e. on T and that

ksN f kye k f ky for all Nf 1 (see [12], Section I.3). Let x A H. By Lemma 4.1 we havethat

��ÐT

�f ðeiyÞ � ðsN f ÞðeiyÞ

�doTðeiyÞx

��2

eÐT

j f ðeiyÞ � ðsN f ÞðeiyÞj2 dðoTÞx;xðeiyÞ:

By dominated convergence the integral on the right-hand side tends to 0 as N ! y. Weconclude that

ÐT


ÐT

ðsN f ÞðeiyÞ doTðeiyÞ in LðHÞ

with convergence in the strong operator topology in LðHÞ. A computation shows that

ÐT

ðsN f ÞðeiyÞ doTðeiyÞ ¼P

jkjeN

1 � jkjN þ 1

� �ff ðkÞTðkÞ in LðHÞ:

This completes the proof of the theorem. r

Remark 4.1. We remark that the argument in the proof of Theorem 4.1 gives that

limk!y

ÐT

fkðeiyÞ doTðeiyÞ ¼ÐT

f0ðeiyÞ doTðeiyÞ in LðHÞð4:2Þ

with convergence in the strong operator topology in LðHÞ whenever f fkg is a boundedsequence of functions in LyðTÞ such that fkðeiyÞ ! f0ðeiyÞ for a.e. eiy A T or fk ! f0 inmeasure. This can be compared with the limit assertion that (4.2) holds in the weak opera-tor topology in LðHÞ whenever fk ! f0 in the weak� topology of LyðTÞ which is evidentfrom the assumption that doT is absolutely continuous with respect to Lebesgue measureon T. By the weak� topology on LyðTÞ we mean the topology on LyðTÞ which is inducedfrom LyðTÞ being the dual of L1ðTÞ.

We mention that the Sz.-Nagy-Foias functional calculus for contractions concernsthe corresponding Abel summability statements for the calculus


uðTÞ ¼ limr!1

Pkf0

akrkT k in LðHÞ;

where u is a bounded analytic function in D with Taylor coe‰cients fakgkf0 (see [19],Chapter III). Our functional calculus gives this calculus as a special case when applied tofunctions f A LyðTÞ with vanishing negative Fourier coe‰cients.

The arguments in this section follow those of Sz.-Nagy and Foias [19], Section III.2,and are repeated here for the matter of convenience.

5. Functional calculus for shift operators

In this section we shall describe the operators T A L�AnðDÞ

�satisfying equality (0.6)

using the functional calculus from Section 4 for the Bergman shift operator Sn on AnðDÞ.We also discuss the functional calculus from Section 4 when applied to the shift operatoron a Hilbert space of analytic functions on the unit disc.

We recall that the operator measure doT from Section 4 is absolutely continuouswith respect to Lebesgue measure on T if T is a completely non-unitary contraction (see[19], Theorem II.6.4). For a contraction T A LðHÞ such that lim

k!yT k ¼ 0 in the strong

operator topology in LðHÞ this assertion of absolute continuity can be seen more easilysince such an operator T can always be modeled as part of a vector-valued adjoint Hardyshift S �

1 (see [19], Subsection I.10.1, or [16] for constructions). See also Proposition 7.2 inSection 7.

Let H be a Hilbert space of analytic functions on the unit disc D. By this we meanthat the elements in H are analytic functions in D and that the point evaluations at pointsin D are continuous linear functionals on H. Associated to such a space H we have a re-producing kernel function KH which is the function KH : D�D ! C uniquely determinedby the properties that KHð�; zÞ A H for every z A D and

f ðzÞ ¼ h f ;KHð�; zÞi; z A D;

for every function f A H. This last property is called the reproducing property of the kernelfunction KH (see [5], Section I.2).

We shall need the following lemma.

Lemma 5.1. Let H be a Hilbert space of analytic functions on D such that the shift

operator S : f 7! zf acts as a contraction on H. Let fckgyk¼�y be a sequence of complex

numbers with limjkj!y

jckj1=jkj e 1 and assume that the limit

T ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ckSðkÞ in LðHÞ

exists in the weak operator topology in LðHÞ. Then the harmonic function


hðzÞ ¼Py

k¼�yckr

jkjeiky; z ¼ reiy A D;ð5:1Þ

is bounded in absolute value by kTk.

Proof. The function h is clearly harmonic in D. We have that

hTKHð�; zÞ;KHð�; zÞið5:2Þ

¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ckhSðkÞKHð�; zÞ;KHð�; zÞi; z A D:

A computation using the reproducing property of the kernel function KH gives that

hSðkÞKHð�; zÞ;KHð�; zÞi ¼ hSkKHð�; zÞ;KHð�; zÞi ¼ zkKHðz; zÞ; z A D;

for kf 0, and similarly that

hSðkÞKHð�; zÞ;KHð�; zÞi ¼ zjkjKHðz; zÞ; z A D;

for k < 0. We write z ¼ reiy. By (5.2) we have that

hTKHð�; zÞ;KHð�; zÞi ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ckr

jkjeikyKHðz; zÞð5:3Þ

¼ hðzÞKHðz; zÞ; z ¼ reiy A D;

since h is harmonic in D. Notice that

jhTKHð�; zÞ;KHð�; zÞije kTk kKHð�; zÞk2 ¼ kTkKHðz; zÞ; z A D;

by the Cauchy-Schwarz inequality and the reproducing property of the kernel function KH.By (5.3) we now conclude that jhðzÞje kTk for all z A D. This completes the proof of thelemma. r

Remark 5.1. We remark that the assumption limjkj!y

jckj1=jkj e 1 in Lemma 5.1 is

redundant in the sense that it follows from the existence of the limit defining the operatorT A LðHÞ. This follows by Theorem 5.2 below and the uniform boundedness principle.

We can now describe the operators T A L�AnðDÞ

�satisfying equality (0.6) using the

functional calculus from Section 4.

Theorem 5.1. Let T A L�AnðDÞ

�be a bounded linear operator. Then the operator T

satisfies equality (0.6) if and only if it has the form of an operator integral

T ¼ÐT


�AnðDÞ

�ð5:4Þ

of a function f A LyðTÞ. Furthermore, we have the norm equality kTk ¼ k f ky.


Proof. By Theorem 3.1 and summability considerations from Section 2 we have thatan operator T A L

�AnðDÞ

�satisfies (0.6) if and only if it has the form of a limit

T ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ckSnðkÞ in L

�AnðDÞ

�ð5:5Þ

with convergence in the strong operator topology in L�AnðDÞ

�for some sequence

fckgyk¼�y of complex numbers. By Theorem 4.1 we conclude that every operator T of theform (5.4) satisfies (0.6). Notice also that kTke k f ky by (1.2) whenever T has the form(5.4).


�be an operator of the form (5.5) with convergence in the

strong operator topology. Notice that jckj ¼ kckSnðkÞke kTk by homogeneity considera-tions from Section 2. By Lemma 5.1 the harmonic function h given by (5.1) is such thatjhðzÞje kTk for all z A D. It is well-known that bounded harmonic functions h in D corre-spond to functions f in LyðTÞ by the Poisson integral formula (see [12], Lemma III.1.2).Passing to boundary values we obtain a function f A LyðTÞ with Fourier coe‰cientsff ðkÞ ¼ ck for k A Z such that k f ky e kTk. By Theorem 4.1 this function f A LyðTÞputs the operator T on the form (5.4) with kTk ¼ k f ky. This completes the proof of thetheorem. r

Let us return to a Hilbert space H of analytic functions on the unit disc D with kernelfunction KH. It is easy to see that the linear span of the reproducing elements KHð�; zÞ,z A D, is dense in H, and that the norm of the function

f ðzÞ ¼Pnj¼1

cjKHðz; zjÞ; z A D;ð5:6Þ

in H, where zj A D for 1e je n, equals

k f k2 ¼Pn

j;k¼1

cjckKHðzk; zjÞð5:7Þ

(see [5], Section I.2).

Proposition 5.1. Let H be a Hilbert space of analytic functions on D such that the

shift operator S : f 7! zf acts as a contraction on H. Then limk!y

S�k ¼ 0 in the strong oper-

ator topology in LðHÞ. In particular, the operator measure doS is absolutely continuous

with respect to Lebesgue measure on T.

Proof. A computation shows that the adjoint shift operator S � acts as

S �KHð�; zÞ ¼ zKHð�; zÞ; z A D;

on reproducing elements. Let f A H be a function of the form (5.6), and notice that

kS�mf k2 ¼Pn

j;k¼1

cjzmj ckz

mk KHðzk; zjÞ ! 0 as m ! y


by (5.7). Since kS�mke 1 for mf 0, we conclude by an approximation argument thatlimm!y

S�m ¼ 0 in the strong operator topology in LðHÞ. The absolute continuity of doS

is now evident by earlier remarks. r

We shall next compute the norm of the operatorÐT

f doS.

Theorem 5.2. Let H be a Hilbert space of analytic functions on D such that the

shift operator S : f 7! zf acts as a contraction on H. Then a measurable function f on T is

integrable with respect to doS if and only if f A LyðTÞ. Furthermore, we have the norm

equality

��ÐT

f ðeiyÞ doSðeiyÞ�� ¼ k f ky; f A LyðTÞ:ð5:8Þ

Proof. Let f A LyðTÞ. Recall that by (1.2) we always have that

�� ÐT

f doS

��e k f ky.By Theorem 4.1 we know that

ÐT

f ðeiyÞ doSðeiyÞ ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ff ðkÞSðkÞ in LðHÞ

with convergence in the strong operator topology in LðHÞ, where ff ðkÞ is the k-th Fouriercoe‰cient of f A LyðTÞ. By Lemma 5.1 we conclude that the Poisson integral h ¼ P½ f �

of f is bounded in absolute value by

�� ÐT

f doT

��. This proves the norm equality (5.8). The

integrability assertion also follows by equality (5.8). This completes the proof of thetheorem. r

We mention that the norm equality (5.8) in Theorem 5.2 also follows by a result ofConway and Ptak [9], Theorem 2.2, using that doS is absolutely continuous with respect toLebesgue measure on T and that the HyðDÞ-functional calculus for the shift operator S isisometric.

Proposition 5.2. Let H be a Hilbert space of analytic functions on D such that the

shift operator S : f 7! zf acts as a contraction on H and let f A LyðTÞ. Then�Ð

T

f ðeiyÞ doSðeiyÞKHð�; zÞ;KHð�; zÞ�

¼ P½ f �ðzÞKHðz; zÞ; z A D;

where P½ f � is the Poisson integral of f .

Proof. By Theorem 4.1 we have that

ÐT

f ðeiyÞ doðeiyÞ ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ff ðkÞSðkÞ in LðHÞ

with convergence in the strong operator topology in LðHÞ. A computation similar to theone in Lemma 5.1 now gives that


�ÐT

f ðeiyÞ doðeiyÞKHð�; zÞ;KHð�; zÞ�

¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ff ðkÞhSðkÞKHð�; zÞ;KHð�; zÞi

¼ P½ f �ðzÞKHðz; zÞ; z A D:

This completes the proof of the proposition. r

We remark that the above results Theorem 5.2 and Proposition 5.2 give converses tothe summability results for the functional calculus in Section 4: If the limit

T ¼ limN!y

PjkjeN

1 � jkjN þ 1

� �ckSðkÞ in LðHÞð5:9Þ

exists in the weak operator topology in LðHÞ, then T ¼ÐT

f doS in LðHÞ for the function

f A LyðTÞ with Fourier coe‰cients ff ðkÞ ¼ ck for k A Z. Similarly, if the limit (5.9) exists inthe uniform operator topology in LðHÞ, then T ¼

ÐT

f doS in LðHÞ with f A CðTÞ. Weomit the details.

We shall next discuss compactness of an operator of the formÐT

f doS. We shall need

the following o¤-diagonal estimate for a reproducing kernel function.

Lemma 5.2. Let H be a Hilbert space of analytic functions on D such that the shift

operator S : f 7! zf acts as a contraction on H and denote by KH the reproducing kernel

function for H. Then

jKHðz; zÞj2 e ð1 � jzj2Þð1 � jzj2Þj1 � zzj2

KHðz; zÞKHðz; zÞ; ðz; zÞ A D�D:

Proof. The essential property needed for a function to be the kernel function for areproducing kernel Hilbert space is that of positive definiteness (see [5], Section I.2). Theassumption that the shift operator S is a contraction on H can equivalently be formulatedsaying that the function

Lðz; zÞ ¼ ð1 � zzÞKHðz; zÞ; ðz; zÞ A D�D;

is positive definite on D�D (see [5], Section I.7). By the Cauchy-Schwarz inequality wehave that

jLðz; zÞjeLðz; zÞ1=2Lðz; zÞ1=2; ðz; zÞ A D�D:

This last inequality gives the conclusion of the lemma. r

We remark that there is a more refined o¤-diagonal estimate for kernel functions re-lated to the operator inequality


kSf þ gk2e 2ðk f k2 þ kSgk2Þ; f ; g A H;

for the shift operator S on H (see [18], Proposition 4.5, and [10], Section 6).

Corollary 5.1. Let H be a Hilbert space of analytic functions on D such that the shift

operator S : f 7! zf acts as a contraction on H. Then the normalized reproducing elements

Kz ¼ KHð�; zÞ=KHðz; zÞ1=2; z A D; KHðz; zÞ3 0;

converge weakly to zero in H as jzj ! 1.

Proof. Notice that hKz;KHð�; zÞi ¼ KHðz; zÞ=KHðz; zÞ1=2, and recall the standardformula

1 � z� z

1 � zz

2

¼ ð1 � jzj2Þð1 � jzj2Þj1 � zzj2

; ðz; zÞ A D�D;

which can be proved by straightforward computation. By the estimate in Lemma 5.2 weconclude that hKz;KHð�; zÞi ! 0 as jzj ! 1 for every z A D. Taking linear combinationswe see that hKz; f i ! 0 as jzj ! 1 for every function f A H of the form (5.6). SincekKzk ¼ 1 for every z A D by construction, the conclusion of the corollary follows by astandard approximation argument. r

We can now show that the operatorÐT

f doS is compact only when f ¼ 0.

Theorem 5.3. Let H be a Hilbert space of analytic functions on D such that the shift

operator S : f 7! zf acts as a contraction on H, and let f A LyðTÞ. Then the operator

ÐT

f ðeiyÞ doSðeiyÞ in LðHÞ

is compact if and only if f ¼ 0.

Proof. It is evident that the zero operator is compact. We turn to the reverse impli-cation. By Proposition 5.2 we have that

�ÐT

f ðeiyÞ doSðeiyÞKz;Kz

�¼ P½ f �ðzÞ; z A D; KHðz; zÞ3 0;

where Kz ¼ KHð�; zÞ=KHðz; zÞ1=2 for z A D, KHðz; zÞ3 0, are the normalized reproducingelements for H and P½ f � is the Poisson integral of f . By Corollary 5.1 we know thatKz ! 0 weakly in H as jzj ! 1. By compactness of the operator

ÐT

f doS we concludethat lim

jzj!1P½ f �ðzÞ ¼ 0. This proves that f ¼ 0. r

6. Toeplitz operators. Harmonic symbols

From the point of view of function theory a Toeplitz operator on the Bergman spaceAnðDÞ, nf 2, is an integral operator Th acting as


ðTh f ÞðzÞ ¼ÐD

1

ð1 � zzÞnhðzÞ f ðzÞ dmnðzÞ; z A D;ð6:1Þ

on, say, polynomials f A AnðDÞ defined using a function h in L1ðD; dmnÞ so that the aboveintegral (6.1) makes sense. The function h is called the symbol for the operator Th, and thefunction

Knðz; zÞ ¼1

ð1 � zzÞn; ðz; zÞ A D�D;

appearing in (6.1) is the reproducing kernel function for the space AnðDÞ. For symbols h

that are harmonic in D the properties of boundedness and compactness of the operator Th

defined by (6.1) are easily settled: The operator Th is bounded as an operator on AnðDÞ ifand only if the symbol h is bounded and kThk ¼ sup

z ADjhðzÞj. The operator Th is compact in

L�AnðDÞ

�if and only if hðzÞ ¼ 0 for all z A D (see [20], Section 6.1).

In the case of the Hardy space A1ðDÞ we can identify a bounded harmonic function h

with its boundary value function in LyðTÞ (see [12], Lemma III.1.2) to make (6.1) a stan-dard formula for the action of a Hardy space Toeplitz operator.

The matrix representation of a Toeplitz operator Th relative to the orthogonal basisof monomials is easily computed.

Lemma 6.1. Let ekðzÞ ¼ zk for kf 0, and let h be a bounded harmonic function in D

with power series expansion

hðzÞ ¼Py

k¼�yckr

jkjeiky; z ¼ reiy A D:ð6:2Þ

Then

hThek; ejiAn¼ cj�kmn; j

for j; kf 0. The matrix representation of the Toeplitz operator Th relative to the orthogonal

basis of monomials fekgkf0 thus has the form fcj�kgj;kf0.

Proof. Notice that the operator Th is obtained by a multiplication by h followed byan orthogonal projection onto AnðDÞ. We have that

hThek; ejiAn¼ÐD

hðzÞzkz j dmnðzÞ ¼ cj�k

ÐD

jzj2j dmnðzÞ ¼ cj�kmn; j

for j; kf 0. This completes the proof of the lemma. r

We can now characterize Toeplitz operators Th with harmonic symbols h using equal-ity (0.6).

Theorem 6.1. Let T A LðHÞ be a bounded linear operator. Then T satisfies equality

(0.6) if and only if T ¼ Th is a Toeplitz operator on AnðDÞ with bounded harmonic symbol h.


Proof. We first show that the m-th homogeneous part ðThÞm of the Toeplitz opera-tor Th with bounded harmonic symbol h given by (6.2) is equal to cmSnðmÞ for m A Z.By Theorem 3.1 this will then show that every such Toeplitz operator Th satisfies equality(0.6).

Let us proceed to details. We first consider the case mf 0. By Lemma 6.1 and con-siderations from Section 2 we have that the m-th homogeneous part ðThÞm of Th is given bythe matrix fdm; j�kcj�kgj;kf0 relative to the orthogonal basis fekgkf0 of monomials. Thismeans that the operator ðThÞm acts as

ðThÞmek ¼ cmemþk ¼ cmSmn ek

for kf 0. We conclude that ðThÞm ¼ cmSmn for mf 0. We consider next the case m < 0.

Notice that ðThÞ�m is the �m-th homogeneous part of T �h ¼ T

h. Since �m > 0, we have by

the case already treated that ðThÞ�m ¼ cmSjmjn . Taking adjoints we see that ðThÞm ¼ cmS

�jmjn

for m < 0.


�be an operator satisfying (0.6). By Theorem 3.1 the m-th ho-

mogeneous part Tm of T has the form Tm ¼ cmSnðmÞ for m A Z, where cm is a complexnumber. Notice also that jcmj ¼ kTmke kTk for m A Z. By considerations from Section 2the operator T can be reconstructed from its homogeneous parts Tm ¼ cmSnðmÞ in thesense that the limit

T ¼ limN!y

PjmjeN

1 � jmjN þ 1

� �cmSnðmÞ in L

�AnðDÞ

�

exists in the strong operator topology in L�AnðDÞ

�. By Lemma 5.1 the harmonic function

h given by (6.2) is bounded in absolute value by kTk. We can now consider the Toeplitzoperator Th in L

�AnðDÞ

�with this harmonic function h as symbol. By the first part of

the proof this Toeplitz operator Th has the same m-th homogeneous part cmSnðmÞ as theoperator T for every m A Z. We conclude that these two operators must be equal: T ¼ Th.This completes the proof of the theorem. r

7. General Toeplitz operators

As we have indicated in the introduction balayage considerations seem to suggest thatBergman space Toeplitz operators should be considered as integrals using naturally associ-ated positive operator measures. We shall indicate in this section some rudiments of such atheory.

Toeplitz operators arise naturally when the functional calculus for normal operatorsis compressed down to a subspace of the original space and the operators obtained in thisway have in many cases properties such as boundedness or compactness for more generalclasses of symbols compared to the original functional calculus for normal operators. Thefunctional calculus for normal operators is defined by integration with respect to the asso-ciated spectral measure and we are led to consider integrability properties of compressedspectral measures.


Let K be a Hilbert space and let dE be an LðKÞ-valued spectral measure on a mea-sure space ðX ;SÞ. Let H be a closed subspace of K and consider the LðHÞ-valued setfunction m defined by

mðSÞ ¼ PHEðSÞjH; S A S;ð7:1Þ

where PH is the orthogonal projection of K onto H. It is apparent that dm is a positiveoperator measure such that mðXÞ ¼ I . Conversely, if we are given a positive operator mea-sure dm on X such that mðXÞ ¼ I , then a construction of Neumark [14] generalizing that ofan L2-space produces a larger Hilbert space K containing H as a closed subspace and anLðKÞ-valued spectral measure dE such that (7.1) holds. This result of Neumark [14] isoften referred to as the Neumark dilation theorem.

A special case often encountered in analysis is when dE is the canonical spectral mea-sure associated to an L2-space. Let dn be a positive (scalar) measure on X , and consider theassociated L2-space L2ðdnÞ. The prototype of a spectral measure dE is given by

EðSÞj ¼ wSj; j A L2ðdnÞ; S A S;

where wS is the characteristic function for the set S. Let now H be a closed subspace ofL2ðdnÞ, and consider the LðHÞ-valued operator measure dmH ¼ dm defined by (7.1). Themarginal distributions dðmHÞj;c of dmH are easily computed as

dðmHÞj;c ¼ jc dn; j;c A H:

We record the following proposition.

Proposition 7.1. Let H be a closed subspace of L2ðdnÞ, let f be a measurable func-

tion on X and set dl ¼ j f j dn. Then f is integrable with respect to dmH if and only if the

embedding ÐX

jjj2 dleCÐX

jjj2 dn; j A H;ð7:2Þ

of H into L2ðdlÞ is continuous. Furthermore, the following statements hold true:

(1) We have the norm bound

�� ÐX

f dmH

��eC with equality if f is non-negative, whereC is the best constant in (7.2).

(2) If the embedding (7.2) is compact, thenÐX

f dmH is a compact operator in LðHÞ.

(3) If f is non-negative andÐX

f dmH is a compact operator in LðHÞ, then the embed-

ding (7.2) is compact.

Sketch of proof. If f is integrable with respect to dmH, thenÐX

jjj2 dl < y for every

j A H, showing that (7.2) holds by the Banach-Steinhaus theorem. Conversely, if (7.2)holds we have by the Cauchy-Schwarz inequality that

ÐX

j f j jjj jcj dne�Ð

X

jjj2j f j dn�1=2�Ð

X

jcj2j f j dn�1=2

eCkjk kck; j;c A H;


showing that f is integrable with respect to dmH, and that

�� ÐX

f dmH

��eC. We notice also

that this last estimation is the best possible forÐX

f jc dm if f is non-negative.

We consider next the compactness properties of Tf ¼ÐX

f dmH. Denote by R the

embedding (7.2) of H into L2ðdlÞ. Let h ¼ sgnð f Þ, where sgnðzÞ ¼ z=jzj for z3 0 andsgnð0Þ ¼ 0, and denote by H the operator on L2ðdlÞ given by multiplication by h. Noticethat the operator Tf ¼

ÐX

f dmH factorizes as

Tf ¼ R�HR in LðHÞ:

This last equality shows that Tf is compact if R is compact. Notice that Tf ¼ R�R if f isnon-negative. If Tf is also compact and jj ! 0 weakly in H, then hTf jj; jji ¼ kRjjk

2 ! 0showing that R is compact. r

The above Proposition 7.1 gives easy criteria for existence as an operator integral,norm bounds and compactness of a generalized Toeplitz operator Tf defined as an LðHÞ-valued operator integral by Tf ¼

ÐX

f dmH. If the space H has bounded point evaluations

H C f 7! f ðxÞ; x A W;

on a set W standard duality arguments give the existence of reproducing elementsKHð�; xÞ A H for x A W such that

f ðxÞ ¼ hj;KHð�; xÞi; x A W;

for j A H. In terms of these reproducing elements the action of the Toeplitz operatorTf ¼

ÐX

f dmH is given by

ðTf jÞðxÞ ¼ hTf j;KHð�; xÞi ¼ÐX

KHð�; xÞ f j dn; x A W;

for j A H.

In complex analysis embeddings of the form (7.2) appear under the name of Carlesonembeddings and in the context of standard weighted Bergman spaces boundedness andcompactness of such embeddings have well-known descriptions using hyperbolic discs orso-called Carleson squares (see [20], Section 6.2). In the context of the Bergman spacesAnðDÞ the natural regularity class for symbols f of Toeplitz operators Tf on AnðDÞ sug-gested here are measurable functions f on D such that j f j dmn is a Bergman space Carlesonmeasure for AnðDÞ. We mention that this regularity class for symbols appears in a recentresult of Miao and Zheng [13] characterizing the compact Toeplitz operators on the un-weighted Bergman space A2ðDÞ in terms of vanishing of the so-called Berezin transform.

A common property possessed by all Bergman shifts is that of being a subnormal op-erator. An operator T A LðHÞ is said to be subnormal if it is part of a normal operatorN A LðKÞ (restriction to an invariant subspace); the operator N A LðKÞ is then called anormal extension of T . Associated to a subnormal operator T A LðHÞ we have the posi-tive LðHÞ-valued operator measure dmT ¼ dm defined by (7.1), where dE is the spectral


measure for a normal extension N A LðKÞ of T . This operator measure dmT is uniquelycharacterized by the property that

T�jT k ¼ÐX

z jzk dmTðzÞ in LðHÞð7:3Þ

for j; kf 0 (see [8], Section II.1).

We recall that the balayage of a complex measure dm on D onto T ¼ qD is the com-plex measure dm 0 on T defined by

ÐT

jðeiyÞ dm 0ðeiyÞ ¼ÐD

P½j�ðzÞ dmðzÞ; j A CðTÞ;

where P½j� is the Poisson integral of j. A straightforward argument using Fubini’s theoremshows that the balayage of dm is given by the L1ðTÞ-function

P�½dm�ðeiyÞ ¼ÐD

Pðz; eiyÞ dmðzÞ; eiy A T;ð7:4Þ

if the mass of dm is carried by the open unit disc D.

We make the following observation.

Proposition 7.2. Let T A LðHÞ be a subnormal contraction and let dmT ¼ dm be the

associated positive operator measure in D given by (7.1) or (7.3). Then

ÐD

uðzÞ dmTðzÞ ¼ÐT

uðeiyÞ doTðeiyÞð7:5Þ

for every function u A CðDÞ which is harmonic in D meaning that doT is the balayage of dmTonto T ¼ qD. In particular, if the mass of dmT is concentrated in the open unit disc D, thendoT is absolutely continuous with respect to Lebesgue measure on T.

Proof. That the operator measures dmT and doT have the same action on harmonicpolynomials is evident by (7.3) and the description of doT in Section 4. Equality (7.5) fol-lows by an approximation argument. By (7.5) the complex measure dðoTÞx;y is the balay-age of dðmTÞx;y onto T ¼ qD giving the last assertion of the proposition. r

We mention that for a more general positive operator measure dm carried by the openunit disc D the operator integral in (7.4) no longer exists in our sense as an operator inLðHÞ.

We wish to mention here also that Toeplitz operators defined as integrals of positiveoperator measures has been studied by Aleman [3], Section III.4, in the context of subnor-mal shift operators in Hilbert spaces of analytic functions; see also Aleman [4].

References

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Department of Mathematics, Mail Stop 942, The University of Toledo, 2801 W. Bancroft St.,

Toledo OH 43606-3390, USA

e-mail: [email protected]

Falugatan 22 1tr, 113 32 Stockholm, Sweden

e-mail: [email protected]

Eingegangen 17. Mai 2006


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