Bergman spaces over the unit ball of C n

sup-ported continuous functions(gj)j such thatkT_g^t−T_g^t_jk−−−→^j→∞ 0. Therefore, by (1.3.18) for any fixedssuch that ₂^t < s < 2twe obtaink˜g^s−ge_j^sk∞ ≤ c(t, s)kT_g^t−T_g^t_jk −−−→^j→∞ 0. Since each ge_j^s ∈C₀(Cⁿ)and is of compact support, then˜g^s∈C₀(Cⁿ).

A converse implication to the above theorem was also proved in [15]. Roughly speaking, if g ∈ T_tsuch that ˜g^s ∈ C₀(Cⁿ)for some0 < s < ₂^t thenT_g^t is compact. The proof uses some known pseudo-differential estimates. More clearly, it is well known [12, 84, 98] that via the Bargmann transform,

β :L²(Rⁿ, dx)−→H_t² defined by[βf](z) = 2ⁿ⁴ R

Rⁿf(x)e

√_π

txz−πx²−^z2

8tdx, every Toeplitz operatorT_g^tonH_t²is unitary equivalent to the Weyl-pseudodifferential operator W

˜

g^t2 acting on L²(Rⁿ, dx). Since g˜^2t ∈ C₀(Cⁿ)an application of Pool´s theorem [143] together with the Calderon-Vaillancourt theorem [84] proves thatW

˜

g^t2 is compact onL²(Rⁿ, dx).

These results together with the equivalence between the compactness ofT_g^tand the vanish-ing at infinity ofg˜^twhenever g is of bounded mean oscillation (c.f. [64]) were used in [15] to show the independence of the compactness of T_g^twith respect to the timet. In Chapter 5, we will use similar techniques to prove that the compactness of Toeplitz operator (with bounded symbols) acting on the Bergman space over a bounded symmetric domain is independent of the standard weights and results similar to Theorems 1.3.1 and 1.3.2 are obtained.

1.4 Bergman spaces over the unit ball of C

ⁿ

The weighted and the ,,unweighted” Bergman spaces over the unit ball ofCⁿand related oper-ators have been studied intensively by several authors [3, 4, 10, 11, 72, 101, 125, 185–187]. In this section, we introduce some properties of these RKHS and compare the situation to the case of the Segal-Bargmann space.

LetΩ =Bⁿbe the open unit ball inCⁿand for anyλ >−1consider the radial weight wλ(z) := cλ(1− |z|²)^λ,

where the normalizing constant c_λ := ^(n+λ)!_πnλ! is chosen so that dµ_λ(z) := w_λ(z)dv(z) is a probability measure on Bⁿ. Theses weights {w_λ}λ>−1 are the standard weights on Bⁿ. In fact, they arise from the Jordan triple determinant polynomial h(z, w) = 1−zwassociated to symmetric domainBⁿ(c.f. Chapter 5).

Denote byA²_λ(Bⁿ)the weighted Bergman space inL²_λ := L²(Bⁿ, dµ_λ). When the weight λ = 0, we simply writeA²(Bⁿ)instead of A²₀(Bⁿ)and refer to A²(Bⁿ)as the ,, unweighted”

Bergman space over the unit ball. The inner product and the norm onL²_λ orA²_λ(Bⁿ)is denoted byh·,·i_λ andk·k_λ, respectively.

36 CHAPTER 1. PRELIMINARIES By Theorem 1.2.1 we know thatA²_λ(Bⁿ)is a reproducing kernel Hilbert space. Moreover, by integration in polar coordinates and using (1.3.5) it is easy to check that the set

(

e_α(z) :=

s

Γ(n+|α|+λ+ 1) α!Γ(n+λ+ 1) z^α

)

α∈Nⁿ

,

whereΓstands for the Gamma function, forms an orthonormal basis ofA²_λ(Bⁿ). Therefore, by (1.2.3) the reproducing kernel ofA²_λ, denoted byK^λ, is given by the series

K^λ(u, z) = X

α∈Nⁿ

Γ(n+|α|+λ+ 1)

α!Γ(n+λ+ 1) u^αz^α = 1

(1−uz)^n+1+λ.

According to (1.2.4) the orthogonal projectionP^λfromL²_λ ontoA²_λ(Bⁿ)is given by the integral formula

[P^λh](z) =hh, K^λ(·, z)i_λ = Z

Bⁿ

h(u)

(1−zu)^n+λ+1dµ_λ(u), h∈L²_λ, z ∈Bⁿ. (1.4.1) Note that, for fixed z ∈ Bⁿ the reproducing kernelK(u, z) is bounded inuand therefore P^λ can be extended toL¹_λ :=L¹(Bⁿ, dµλ)by its integral representation in (1.4.1). Therefore, for a functiong ∈L¹_λ the Toeplitz operatorT_g^λ, given by

T_g^λh(z) = [P^λgh](z) = Z

Bⁿ

h(u)g(u)

(1−zu)^n+λ+1dµλ(u),

is densely defined onA²_λ and its domain of definitionD(T_g^λ)contains all bounded holomorphic functions onBⁿ. SinceP^λ is not bounded onL¹_λthe operatorT_g^λis unbounded in general.

The Berezin transform of a function g ∈ L¹_λ, denoted g˜^λ, is defined to be the Berezin transform of the Toeplitz operatorT_g^λ:

˜

g^λ(z) :=hT_g^λk_z^λ, k_z^λi=hgk^λ_z, k_z^λi= (1− |z|²)^n+λ+1 Z

Bⁿ

g(u)

|1−uz|^2(n+λ+1)dµ_λ(u), (1.4.2) wherek^λ_z is the normalizing kernel function given byk_z^λ(u) = (1−|z|²)^12(n+λ+1) 1

(1−uz)^n+1+λ. Note that, by the above definition the Berezin transformg˜^λ replaces the heat transform in the case of the Segal-Bargmann space. Since the mapping g −→ T_g^λ is bounded on L^∞(Bⁿ) it follows that the Berezin transformB_λ :g −→g˜^λ is bounded onL^∞(Bⁿ)withk˜g^λk∞ ≤ kgk∞. Moreover, in [125] it has been shown that the mappingB_λ is bounded onL^p(Bⁿ, dµ_λ)if and only ifp >1. Note thatB_λis one-to-one onL¹_λ (c.f. p. 32 in [101] for the casen = 1). Indeed, suppose˜g^λ = 0for someg ∈L¹_λ then the function

G(z) := g˜^λ(z)

(1− |z|²)^n+λ+1 = Z

Bⁿ

g(u)

(1−uz)^(n+λ+1)(1−uz)^(n+λ+1)dµλ(u) = 0.

Therefore, for any α, β ∈ Nⁿ0 we have (∂_z^α∂_z^βG)(0) = 0 , where ∂_z^α := ∂_z^α₁¹∂_z^α₂²· · ·∂_z^α_nⁿ. Dif-ferentiation under the integral sign shows that R

Bⁿu^αu^βg(u)dµ_λ(u) = 0henceg = 0 a.e. on Bⁿ.

1.4. BERGMAN SPACES OVER THE UNIT BALL OFC^N 37 There is another useful way to write the Berezin transform as an integral involving the involutive automorphismsϕ_z of the ball. For eachz ∈Bⁿthe functionϕ_z is the automorphism Bⁿwhich exchanges0andzand is given by

ϕ_z(u) = z−P_z(u)−p

1− |z|²Q(u)

1−uz u∈Bⁿ,

whereP_z is the orthogonal projection fromCⁿ on to the one-dimensional subspace generated by z and Q(u) = u −P_z(u). Remark that the automorphism group of the ball Aut(Bⁿ) is generated by the unitary operators onCⁿ and the involutions{ϕ_z}z∈Bⁿ (c.f. Theorem 2.2.5 in [149]). Moreover, we have

1− |ϕ_z(u)|² = (1− |z|²)(1− |z|²)

|1−uz|²) , (1.4.3)

and

JRϕz(u) =

1− |z|²

|1−uz|² n+1

, whereJRϕzis the real Jacobian ofϕz. (1.4.4) Therefore, by the change of variableu−→ϕ_z(u)together with (1.4.3) and (1.4.4) we obtain:

˜ g^λ(z) =

Z

Bⁿ

g(u)(1− |z|²)^n+λ+1

|1−uz|^2(n+λ+1) dµ_λ(u)

= Z

Bⁿ

g◦ϕ_z(u)dµ_λ(u). (1.4.5)

Let us choose a functiong ∈L¹_λfor all−1< λ≤0such that the Berezin transform satisfies the semigroup property (g˜e^νs = ˜g^ν+s). Then by (1.4.5) we have

˜e

g^λ0(z) = Z

Bⁿ

˜

g^λ◦ϕ_z(u)dµ_λ(u) = ˜g^λ(z) = Z

Bⁿ

g◦ϕ_z(u)dµ_λ(u).

Since the Berezin transform map is one-to-one onL¹_λ we see that˜ge^λ0 = ˜g^λif and only ifg = ˜g^λ (compare the situation to the Segal-Bargmann case). Therefore, it is natural to study the fix points of the Berezin transform. In fact, the problem of characterizing the functions invariant under the Berezin transform has been encountered by several authors [2, 8, 75, 137]. In the case n = 1, and under the assumption thatg is continuous up to the boundaryg is a fix point of the Berezin transform if and only if it is a harmonic function (c.f. Prop. 6.20 in [186]). This result is still true if we assume only that g is bounded on the open unit disc [75].

We aim now to prove that the M-harmonic functions onBⁿare fixed points of the Berezin transform. Recall that a twice differentiable function g is called M-harmonic if ( ˜∆g)(z) :=

∆(g◦ϕ_z)(0) = 0for allz ∈ Bⁿ. In casen = 1, they are precisely the harmonic functions and this fails to be true forn > 1(c.f. Remark 4.1.4 in [149]).

Letgbe M-harmonic function then by the mean value property of the M-harmonic functions (Theorem 4.1.3 in [149]) we can write

38 CHAPTER 1. PRELIMINARIES

g(ϕ_z(0)) = 1 σ(S²ⁿ⁻¹)

Z

S²ⁿ⁻¹

g(ϕ_z(rξ))dσ(ξ), for every0< r <1.

Multiplying both sides of the above equation by a factor of 2nr²ⁿ⁻¹(1−r²)^λ and integrating over[0,1)we obtain

2ng(z) Z 1

0

r²ⁿ⁻¹(1−r²)^λdr= 1 σ(S²ⁿ⁻¹)

Z

Bⁿ

g◦ϕ_z(u)(1− |u|²)dv(u).

Therefore,

g(z) = Z

Bⁿ

g◦ϕ_z(u)dµ_λ(u) = ˜g^λ(z).

By the above calculations it should be noted that any functiong ∈ L¹_λ is a fix point for the Berezin transformB_λif and only ifgsatisfies the mean value property . In case ofλ= 0, it has been proved in [2] that functions inL¹₀ which satisfy the mean value property are exactly those M-harmonic in casen ≤11and this fails for higher dimensions. Therefore there are fix points of the Berezin transformB₀which are not M-harmonic, and according to Proposition 13.4.4 in [149] these functions do not admit continuous extension to the closed ballBⁿ.

Similar to the case of the Segal-Bargmann space it is clear that ifg ∈ L¹_λ thenk˜g^λ0k∞ ≤ k˜g^λk∞ ≤ kT_g^λk for allλ₀ ≥ λ. Therefore, if T_g^λ is bounded then the Berezin transformg˜^λ0 is bounded for allλ₀ ≥ λ. Moreover, sincek_z^λ −−−−→^z→∂Bn 0 weakly˜g^λ0 ∈ C₀(Bⁿ)whenever T_g^λ is compact for some λ ≤ λ₀. Furthermore, if we suppose that g is M-harmonic then T_g^λ is bounded (respectively compact) if and only ifg is bounded (respectivelyg = 0). Indeed, since the M-harmonic functions are fix points of the Berezin transformT_g^λ is bounded if and only if g = ˜g^λ is bounded. Now, if we suppose thatT_g^λ is compact theng = ˜g^{λ z}−−−−→^→∂Bn 0and therefore by the maximum principle of the M-harmonic functions (c.f. Theorem 4.3.2 in [149]) it follows thatg = 0.

The connection between the vanishing of the Berezin transform of the function near the boundary and the compactness of its corresponding Toeplitz operator has been studied fre-quently [10, 158, 183, 187]. In [187] it was proved that for symbolsgof bounded mean oscilla-tiong˜⁰ ∈C₀(Bⁿ)if and only ifT_g⁰is compact onA²(Bⁿ). Later on in [15] and as an analogue to the Segal-Bargmann space case, the existence of a universal constantc(λ, λ₀)was shown such that

k˜g^λk_∞ ≤c(λ, λ₀)kT_g^λk,

forgin a suitable class of symbols. This estimation together with the result [187] were used in [15] to prove that the compactness of the Toeplitz operator T_g^λ is independent of the weightλ whenevergis of bounded mean oscillation.

Chapter 2

Commuting Toeplitz operators with quasi-homogeneous symbols

on the Segal-Bargmann space

For the Gaussian measure µ on the complex plane C given by the density dµ := dµ¹

4 =

1

πe^−|·|2dv, where v is the usual Lebesgue measure, we denote by H² := H²1 4

= H²(C, dµ) the Segal-Bargmann space over C. As introduced in the previous chapterH² is a closed sub-space ofL² :=L²(C, dµ)consisting of allµ-square integrable entire functions onC. We know thatH² is a RKHS with the reproducing kernelK(w, z) = e^wz. In particular, ifP denotes the orthogonal projection ofL² ontoH² then

[P h](z) =hh, K(·, z)i= Z

C

h(w)e^zwdµ(w), ∀z ∈C, whereh·,·idenotes the usual inner product onL².

For a functiong ∈ E := E¹

2 ⊂ E¹

4 (c.f. Definition 1.3.1), the Toeplitz operatorT_g :=T

1

g4 is given by

T_g :D(T_g) :=n

h∈H² |gh∈L²(C, dµ)o

⊂H² −→H² :h7→P(gh).

According to Proposition 1.3.1 the operatorT_g is densely defined andD(T_g)contains the space of holomorphic polynomialsP[z]as well as the linear space

L:=L_H² = span{k_z(w) := K(w, z)e^−12|z|2}z∈C.

In this chapter we are motivated by the following problem: Let T = T_zlz^k with l, k ∈ N0 be a Toeplitz operator with monomial symbol acting on H². Determine the symbols Ψ of polynomial growth at infinity such that T_Ψ and T_zlz^k commute on the space of all holo-morphic polynomials in H². By using polar coordinates we represent Ψ as an infinite sum Ψ(re^iθ) = P∞

j=−∞Ψ_j(r)e^ijθ. Then we are able to reduce the above problem to the case of quasi-homogeneous symbolsΨ = Ψ_je^ijθ. We obtain the radial partΨ_j(r)in terms of the inverse

39

40 CHAPTER 2. COMMUTING TOEPLITZ OPERATORS Mellin transform of an expression which is a product of Gamma functions and a trigonometric polynomial. If we allow operator symbols of higher growth at infinity (but remaining inE), we point out that in some of the cases more than one Toeplitz operator T_Ψje^ijθ exists commuting withT.

The following section describes this problem in details as well as the steps used for solving it. We also state two of our main results which we prove throughout the chapter.

2.1 Introduction

The problem of characterizing the symbols of commuting Toeplitz operators on (weighted) Bergman spaces over various domains as well as the study of algebras of commuting Toeplitz operators, has attracted the interest of several authors [8, 9, 16, 17, 60, 61, 66, 126, 129, 130, 144, 168, 169, 172]. The analysis often is restricted to the case where at least one of the symbols belongs to a certain subclass of functions.

On the Hardy spaceH²(S¹), Brown and Halmos [54] were the first to obtain a complete de-scription of bounded symbols so that the corresponding Toeplitz operators commute onH²(S¹).

In fact, they proved that two Toeplitz operators with bounded symbols commute on H²(S¹)if and only if both symbols are analytic, or both are conjugate analytic, or one of them is a linear function of the other. An analogous problem in the case of the Bergman space A(D)over the unit discDwas considered by S. Axler and Z. ˘Cu˘ckovi´c in [8]. Roughly speaking, they showed that for a pair of commuting Toeplitz operators onA(D)with bounded harmonic symbols the result obtained by Brown and Halmos in [54] is still true. Of course, harmonicity is essential since any two Toeplitz operators with radial symbols commute on A(D). Later on, with Rao [9], they proved that if two Toeplitz operators on A(D)commute and ,, the symbol of one of them is non constant and analytic then the other is analytic too” [9].

Letϕbe a monomial or more generally a bounded quasi-homogeneous function on D. In [66, 130] a complete characterization of the Toeplitz operatorT_ψ onA(D)with bounded symbol ψ was given such that T_ψ commutes with T_ϕ. More precisely, every bounded functionψ can be represented by anL²(D, dv)-convergent seriesψ(re^iθ) =P∞

j=−∞ψ_j(r)e^ijθ whereψ_j(r)are bounded functions defined on the interval[0,1)( herer andθ represent the polar coordinated inC). Using this fact, Z. ˘Cu˘ckovi´c and N. V. Rao deduced thatT_ψ commutes withT_ϕcommute if and only if for each j ∈ Z the Toeplitz operator T_ψj(r)eijθ commutes with T_ϕ. Then they were able to describe each functionψ_j as the inverse Mellin transform of a product of Gamma functions (c.f. Theorem A0).

An analogous problem in the case of the Segal-Bargmann space and considering ψ to be of polynomial growth at infinity is investigated in this chapter (for a detailed description of the problem c.f. below). In order to compare our results to those in the case of the Bergman space over the unit disc and for the sake of completeness we state the main result in [66].

Theorem A0. [66] Let ψ(re^iθ) = P∞

j=−∞ψ_j(r)e^ijθ and ϕ(re^iθ) = r^me^iδθ be bounded func-tions on D, where ψ_j are bounded, m ∈ N0 andδ ∈ N. ThenT_ϕT_ψ = T_ψT_ϕ onA(D)if and

2.1. INTRODUCTION 41 only if for eachj ∈Zthere is a constanta_j such that

ψ_j(r) = a_jM⁻¹

"

Γ(^z+j_2δ )Γ(^z+m+δ−j_2δ ) Γ(^z+2δ−j_2δ )Γ(^z+m+δ+j_2δ )

#

(r), (2.1.1)

whereM⁻¹denotes the inverse Mellin transform.

Moreover, it was shown in [66] that for bounded functions Φ₀ and Φ, where Φ₀ is non-constant and radial (i.e. ϕ(re^iθ) =ϕ(r)), the operatorsT_Φ0 andT_Φ commute if and only ifΦis radial.

In the case of the Segal-Bargmann space it was recently shown in [16, 17] that the growth of symbols near infinity essentially influences the results. On the one hand, if T_ϕ1 and T_ϕ2 commute such that both symbols are of polynomial growth and one of them is radial non-constant, then the other symbol must be radial, too. On the other hand, examples of commuting operatorsT_ϕ1 andT_ϕ2 exist whereϕ₁ is radial of exponential growth at infinity andϕ₂ is non-radial and bounded, c.f. Example 2.7.3 of Section 7 and Example 5.6 in [17].

In this chapter, we fix a monomialfm,δ(re^iθ) = r^me^iδθ, wherem=l+kandδ =l−kwith l, k ∈ N⁰. We search for functionsΨof polynomial growth at infinity such that Tf_m,δ andTΨ

commute as operators on the space of holomorphic polynomials. Analogous to the case of the Bergman space over the unit disc, we expressΨin a form of anL²-convergent series of quasi-homogeneous functionsΨ(re^iθ) = P∞

j=−∞Ψj(r)e^ijθ. It turns out, as in the case considered by Z. ˘Cu˘ckovi´c and N. V. Rao in [66], that it is sufficient to consider this problem for the operators Tf_m,δ andT_Ψje^ijθ. More precisely:

(1) Under the assumption[T_fm,δ, T_Ψje^ijθ] := T_fm,δT_Ψje^ijθ −T_Ψje^ijθT_fm,δ = 0, we characterize the functionsΨ_j as the inverse Mellin transform of an expression formed up of Gamma functions and a trigonometric polynomial.

(2) For each fixedj ∈ Z, we give a collection of quasi-homogeneous symbols Ψ_je^ijθ such that the commutator[T_fm,δ, T_Ψje^ijθ]vanishes.

The main idea in (1) is to derive a functional equation for the Mellin transform ofΨ_je^−r2 on some right half plane together with additional conditions starting from the relations

[Tf_m,δ, T_Ψje^ijθ](z^k) = 0, for all k ∈N⁰.

We then, construct all possible solutions under the assumption thatΨhas polynomial growth at infinity. Finally, we obtainΨ_j via the inverse Mellin transform.

In (2) we use the symbols obtained in (1) and study their growth behavior, to find the can-didates of quasi-homogeneous symbols Ψ_je^ijθ such that [T_fm,δ, T_Ψje^ijθ] = 0. We prove that, for j ∈ Z sufficiently large, there exists at least one symbolΨ_j with this property having at

42 CHAPTER 2. COMMUTING TOEPLITZ OPERATORS most polynomial growth at infinity. Moreover, we point out that in general there are additional symbolsΨ_j of exponential growth such thatT_fm,δ andT_Ψje^ijθ commute.

We show that it is sufficient to treat only the caseδ > 0. Then we decompose the problem into three parts: (1) casej > δ(2) casej <0, (3)06j 6δ.

Now we state two of our main results. The first one addresses the problem (1)

Theorem A1. Let Ψbe a measurable complex valued function on Cof polynomial growth at infinity. We writeΨ(re^iθ) =P∞

j=−∞Ψj(r)e^ijθ as an expansion inL²(C, dµ). For eachj > δ, we define a holomorphic functionGj(z)forRe(z)>−j−1by

Gj(z) :=

j

Y

l=1

Γ(z+p+l

δ )

j−δ

Y

l=1

Γ(z+δ+l

δ )

−1

δ^z,

where p := ^δ+m₂ . Suppose that T_fm,δT_Ψ = T_ΨT_fm,δ. Then for each j > δ there exists a trigonometric polynomialp(z) = P

|l|<^δ

4

a_le^2πilzδ such that

Ψ_j(r) = 2M⁻¹[G_j(z)p(z)] (r²)r^−j−2e^r2, whereM⁻¹denotes the inverse Mellin transform.

On the other hand we have:

Theorem A2. Letf_m,δ(re^iθ) = r^me^iδθ be a monomial, and letl∈Zsuch that|l|< _2π^δ arccos³₄. Then for eachj > δthe function

ϕ_j(r)e^ijθ :=M⁻¹h

G_j(z)e^2πilzδ i

(r²)r^−j−2e^r2e^ijθ,

defines an operator symbol such that the commutator[Tf_m,δ, T_ϕje^ijθ]is well defined and vanish-ing on the space of holomorphic polynomials. Moreover,ϕj(r)e^ijθ is of polynomial growth at infinity in casel = 0.

Results analogous to Theorems A1 and A2 and in the casesj < 0and0 6 j 6 δ are also obtained (c.f. Theorems 2.5.1-2.6.1 and Corollary 2.6.1).

The chapter is organized as follows. In Section 2.2 we setup the notations and give some standard results used in our work. In particular, we make clear that the operator products T_fm,δT_Ψje^ijθ and T_Ψje^ijθT_fm,δ are well defined on a dense subset of H² containing the holo-morphic polynomials and we reduce the problem to the case δ ≥ 0. Section 2.3 is devoted to the proof that [T_fm,δ, T_Ψ] = 0if and only if [T_fm,δ, T_Ψje^ijθ] = 0for all j ∈ Z. Moreover, we give further equivalent conditions involving the Mellin transform of each {Ψ_j}j∈Z at specific points. In Section 2.4 we consider the case j > δ and under the assumption that Ψis of at most polynomial growth at infinity and [T_fm,δ, T_Ψje^ijθ] = 0, we derive a functional equation for the Mellin transform ofΨ_je^−r2 on some right half plane. Moreover, for an infinite number

2.2. PRELIMINARIES 43

Im Dokument The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions. (Seite 39-47)