characterizations of bergman space toeplitz operators with ...ao/lo2008.pdf · characterizations of...
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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 Sn�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
4 Louhichi and Olofsson, Bergman space Toeplitz operators
(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
f ðeiyÞ doSnðeiyÞ in L
�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.
5Louhichi and Olofsson, Bergman space Toeplitz operators
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 Sn�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
n�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
6 Louhichi and Olofsson, Bergman space Toeplitz operators
�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
7Louhichi and Olofsson, Bergman space Toeplitz operators
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;
8 Louhichi and Olofsson, Bergman space Toeplitz operators
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 Sn�1S �n S
kn SnðS �
n SnÞ�1 ¼ Skn ðS �
n Sn�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 .
9Louhichi and Olofsson, Bergman space Toeplitz operators
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:
10 Louhichi and Olofsson, Bergman space Toeplitz operators
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
11Louhichi and Olofsson, Bergman space Toeplitz operators
ðS 0nÞ
�te�iyTte iyS0n ¼ te�iyðS 0
n�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:
�
12 Louhichi and Olofsson, Bergman space Toeplitz operators
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
f ðeiyÞ doTðeiyÞ ¼ limN!y
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
13Louhichi and Olofsson, Bergman space Toeplitz operators
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
f ðeiyÞ doTðeiyÞ ¼ limN!y
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
f ðeiyÞ doTðeiyÞ ¼ limN!y
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
14 Louhichi and Olofsson, Bergman space Toeplitz operators
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
15Louhichi and Olofsson, Bergman space Toeplitz operators
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
f ðeiyÞ doSnðeiyÞ in L
�AnðDÞ
�ð5:4Þ
of a function f A LyðTÞ. Furthermore, we have the norm equality kTk ¼ k f ky.
16 Louhichi and Olofsson, Bergman space Toeplitz operators
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).
Let now T A L�AnðDÞ
�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
17Louhichi and Olofsson, Bergman space Toeplitz operators
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
18 Louhichi and Olofsson, Bergman space Toeplitz operators
�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
19Louhichi and Olofsson, Bergman space Toeplitz operators
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
20 Louhichi and Olofsson, Bergman space Toeplitz operators
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.
21Louhichi and Olofsson, Bergman space Toeplitz operators
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.
Let now T A L�AnðDÞ
�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.
22 Louhichi and Olofsson, Bergman space Toeplitz operators
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;
23Louhichi and Olofsson, Bergman space Toeplitz operators
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
24 Louhichi and Olofsson, Bergman space Toeplitz operators
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
[1] P. Ahern, On the range of the Berezin transform, J. Funct. Anal. 215 (2004), 206–216.
[2] P. Ahern and Z. Cuckovic, A theorem of Brown-Halmos type for Bergman space Toeplitz operators, J.
Funct. Anal. 187 (2001), 200–210.
25Louhichi and Olofsson, Bergman space Toeplitz operators
[3] A. Aleman, The multiplication operators on Hilbert spaces of analytic functions, Habilitationsschrift, Fern-
universitat Hagen, 1993.
[4] A. Aleman, Subnormal operators with compact selfcommutator, Manuscr. Math. 91 (1996), 353–367.
[5] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
[6] S. Axler and Z. Cuckovic, Commuting Toeplitz operators with harmonic symbols, Integr. Equ. Oper. Th. 14
(1991), 1–12.
[7] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. reine angew. Math. 213 (1963/
1964), 89–102.
[8] J. B. Conway, The theory of subnormal operators, American Mathematical Society, 1991.
[9] J. B. Conway and M. Ptak, The harmonic functional calculus and hyperreflexivity, Pacific J. Math. 204
(2002), 19–29.
[10] H. Hedenmalm, S. Jakobsson, and S. M. Shimorin, A biharmonic maximum principle for hyperbolic surfaces,
J. reine angew. Math. 550 (2002), 25–75.
[11] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Springer, 2000.
[12] Y. Katznelson, An introduction to harmonic analysis, Dover Publications, 1976.
[13] J. Miao and D. Zheng, Compact operators on Bergman spaces, Integr. Equ. Oper. Th. 48 (2004), 61–79.
[14] M. A. Neumark, On a representation of additive operator set functions, C. R. (Doklady) Acad. Sci. URSS
(N.S.) 41 (1943), 359–361.
[15] A. Olofsson, Operator-valued n-harmonic measure in the polydisc, Studia Math. 163 (2004), 203–216.
[16] A. Olofsson, An operator-valued Berezin transform and the class of n-hypercontractions, Integr. Equ. Oper.
Th. 58 (2007), 503–549.
[17] A. Olofsson, A characteristic operator function for the class of n-hypercontractions, J. Funct. Anal. 236
(2006), 517–545.
[18] S. M. Shimorin, Wold-type decompositions and wandering subspaces of operators close to isometries, J. reine
angew. Math. 531 (2001), 147–189.
[19] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, 1970.
[20] K. Zhu, Operator theory in function spaces, Marcel Dekker, 1990.
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
26 Louhichi and Olofsson, Bergman space Toeplitz operators