# Schatten Class Toeplitz Operators on the Bergman Space

Authors: Namita Das
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We have shown that if the Toeplitz operatorTϕ on the
Bergman space La2(𝔻) belongs to the Schatten class Sp,1≤p<∞, then ϕ˜∈Lp(𝔻,dλ), where ϕ˜ is the Berezin transform of ϕ,dλ(z)=dA(z)/(1−|z|2)2, and dA(z) is the normalized area measure on the open unit disk 𝔻. Further, if ϕ∈Lp(𝔻,dλ) then ϕ˜∈Lp(𝔻,dλ) and Tϕ∈Sp. For certain
subclasses of L∞(𝔻), necessary and sufficient conditions characterizing
Schatten class Toeplitz operators are also obtained.


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## Body

## 1. Introduction

Let𝔻={z∈ℂ:|z|<1} be the open unit disc in the complex plane ℂ. Let dA(z)=(1/π)dxdy=(1/π)rdrdθ and L2(𝔻,dA) be the Hilbert space of complex-valued, absolutely square-integrable, Lebesgue measurable functions f on 𝔻 with the inner product
(1.1)〈f,g〉=∫f(z)g(z)¯dA(z).
Let L∞(𝔻,dA) denote the Banach space of Lebesgue measurable functions f on 𝔻 with
(1.2)∥f∥∞=esssup{|f(z)|:z∈𝔻}<∞,
and let H∞(𝔻) be the space of bounded analytic functions on 𝔻. Let La2(𝔻) be the subspace of L2(𝔻,dA) consisting of analytic functions. The space La2(𝔻) is called the Bergman space. Since point evaluation at z∈𝔻 is a bounded linear functional on the Hilbert space La2(𝔻), the Riesz representation theorem implies that there exists a unique function Kz in La2(𝔻) such that
(1.3)f(z)=∫𝔻f(w)Kz(w)¯dA(w)
for all f in La2(𝔻). Let K(z,w) be the function on 𝔻×𝔻 defined by
(1.4)K(z,w)=Kz(w)¯.
The function K(z,w) is thus the reproducing kernel for the Bergman space La2(𝔻) and is called the Bergman kernel. It can be shown that the sequence of functions {en(z)}={n+1zn}n≥0 forms the standard orthonormal basis for La2(𝔻) and K(z,w)=∑n=1∞en(z)en(w)¯. The Bergman kernel is independent of the choice of orthonormal basis and K(z,w)=1/(1-zw̅)2.Letka(z)=K(z,a̅)/K(a̅,a̅)=(1-|a|2)/(1-a̅z)2. These functions ka are called the normalized reproducing kernels of La2; it is clear that they are unit vectors in La2. For any a∈𝔻, let ϕa be the analytic mapping on 𝔻 defined by ϕa(z)=(a-z)/(1-a̅z), z∈𝔻. An easy calculation shows that the derivative of ϕa at z is equal to -ka(z). It follows that the real Jacobian determinant of ϕa at z is
(1.5)Jϕa(z)=|ka(z)|2=(1-|a|2)2|1-a̅z|4.
When |z|→1, kz→0 weakly and the normalized reproducing kernels kz, z∈𝔻 span La2(𝔻) [1]. Since La2(𝔻) is a closed subspace of L2(𝔻,dA) (see [1]), there exists an orthogonal projection P from L2(𝔻,dA) onto La2(𝔻). For ϕ∈L∞(𝔻), we define the Toeplitz operator Tϕ on La2(𝔻) by Tϕf=P(ϕf), f∈La2(𝔻). For ϕ∈L∞(𝔻), let ϕ̃(z)=〈Tϕkz,kz〉. The function ϕ̃(z) is called the Berezin transform of ϕ. Let Hϕ be the Hankel operator from La2 into (La2)⊥ defined by Hϕf=(I-P)(ϕf). It is easy to check that Hϕ*Hϕ=T|ϕ|2-Tϕ¯Tϕ (see [1]). For ψ∈L∞(𝔻), define Sψ:La2→La2 as Sψf=PJ(ψf), where J:L2→L2 is defined as Jf(z)=f(z̅). The operator Sψ is called the little Hankel operator on La2(𝔻). Let dλ(z)=K(z,z)dA(z)=dA(z)/(1-|z|2)2, the Mobius invariant measure on 𝔻. Let ℒ(H) be the set of all bounded linear operators from the Hilbert space H into itself, and let ℒ𝒞(H) be the set of all compact operators in ℒ(H).Often it is not easy to verify that a linear operator is bounded, and it is even more difficult to determine its norm. No conditions on the matrix entriesaij have been found which are necessary and sufficient for A to be bounded, nor has ∥A∥ been determined in the general case. For the more general problem we also need analogues of the notions of operator norm. For more details see [2, 3]. The family of norms that has received much attention during the last decade is the Schatten norm.A proper two-sided ideal𝒯 in ℒ(H) is said to be a norm ideal if there is a norm on 𝒯 satisfying the following properties:(i)
(𝒯,∥·∥𝒯) is a Banach space;(ii)
∥AXB∥𝒯≤∥A∥∥X∥𝒯∥B∥ for all A,B∈ℒ(H) and for all X∈𝒯;(iii)
∥X∥𝒯=∥X∥ for X a rank one operator.If(𝒯,∥·∥𝒯) is a norm ideal, then the norm ∥·∥𝒯 is unitarily invariant, in the sense that ∥UAV∥𝒯=∥A∥𝒯 for all A in 𝒯 and unitary U,V in ℒ(H). Each proper ideal of ℒ(H) is contained in the ideal of compact operators. Two special families of unitarily invariant norms satisfying conditions (i), (ii), and (iii) are the Schatten p-norms defined on the set of compact operators and the Ky Fan norms.For any nonnegative integern, the nth singular value of T∈ℒ𝒞(H) is defined by
(1.6)sn(T)=inf{∥T-K∥:K∈ℒ(H),rankK≤n}.
Here ∥·∥ is the operator norm. Clearly, s0(T)=∥T∥ and
(1.7)s0(T)≥s1(T)≥s2(T)≥⋯≥0.Thussn(T) is the distance, with respect to the operator norm, of T from the set of operators of rank at most n in ℒ(H). The spectral theorem shows that the singular values of the compact operator T are the square roots of the eigenvalues of T*T as long as H is separable and infinite dimensional. Notice that sn(T) can be defined for any T∈ℒ(H) but clearly, sn(T)→0 if and only if T is compact.The Schatten Von Neumann classSp=Sp(H), 0<p<∞, consists of all operators T∈ℒ𝒞(H) such that
(1.8)∥T∥Sp=(∑n=0∞(sn(T))p)1/p<∞.If1≤p<∞, then ∥·∥Sp is a norm, which makes Sp a Banach space. For p<1, ∥·∥Sp does not satisfy the triangle inequality, it is a quasinorm (i.e., ∥T1+T2∥Sp≤C(∥T1∥Sp+∥T2∥Sp) for T1,T2∈Sp and C, a constant), which makes Sp a quasiBanach space. We will be mainly concerned with the range 1≤p<∞. The space S1 is also called the trace class and S2 is called the Hilbert-Schmidt class. The linear functional trace is defined on S1 by
(1.9)traceT=∑n=0∞〈Tεn,εn〉,T∈S1,
where {εn}n≥0 is an orthonormal basis in H. Moreover, the right-hand side does not depend on the choice of the basis. If 1<p<∞, the dual space Sp* can be identified with Sq with respect to the pairing 〈T,R〉=traceTR*, T∈Sp, R∈Sq. Here q=p/(p-1) is the dual exponent. With respect to the same pairing one can identify S1* with ℒ(H) and (ℒ𝒞(H))* with S1. We refer the reader to [2, 3] for basic facts about Schatten p-classes. The Schatten p-classes should be seen as gradations of compactness for an operator. Each Schatten p-class is dense in the space of compact operators in the operator norm. For this reason, it is of interest, given a certain class of operators, to ask whether or not there are compact operators not in any Schatten p-class. For instance, this was proved by Arazy et al. [4] and Zhu [1] for Hankel operators on Bergman space.Hankel operators are closely related to Toeplitz operators. Many problems about Toeplitz operators can also be formulated in terms Hankel operators and vice versa. The singular values of Hankel operators on the Hardy space play a crucial role in rational approximation. The celebrated results of Adamjan et al. [5] which give the achievable error in approximating a Hankel operator Γ by another one of smaller rank in terms of the singular values of the Hankel operator is an illustration of this. It may be noted here that the Adamjan, Arov, and Krein theorem has had a considerable influence on the treatment of the problem that arises in engineering applications in the context of model reduction, that is, the problem of finding a simple model to replace a relatively complicated one without too great a loss of accuracy. In view of this it would be nice to have a satisfactory characterization for Schatten class Toeplitz operators on the Bergman space.In this paper we find necessary and sufficient conditions onϕ that will ensure that the Toeplitz operator Tϕ belong to Sp, 1≤p<+∞. This will provide some quantitative estimates (size estimates of these operators) in terms of norms. We will also use S∞ to denote the full algebra of bounded linear operators from the Bergman space La2(𝔻) into itself.Forz and w in 𝔻, let ϕz(w)=(z-w)/(1-z̅w). These are involutive Mobius transformations on D. In fact(1)
ϕz∘ϕz(w)≡w;(2)
ϕz(0)=z, ϕz(z)=0;(3)
ϕz has a unique fixed point in 𝔻.Givenz∈𝔻 and f any measurable function on 𝔻, we define a function Uzf(w)=kz(w)f(ϕz(w)). Since |kz|2 is the real Jacobian determinant of the mapping ϕz (see [1]), Uz is easily seen to be a unitary operator on L2(𝔻,dA) and La2(𝔻). It is also easy to check that Uz*=Uz, thus Uz is a self-adjoint unitary operator. If ϕ∈L∞(𝔻,dA) and z∈𝔻 then UzTϕ=Tϕ∘ϕzUz. This is because PUz=UzP and for f∈La2, Tϕ∘ϕzUzf=Tϕ∘ϕz((f∘ϕz)kz)=P((ϕ∘ϕz)(f∘ϕz)kz)=P(Uz(ϕf))=UzP(ϕf)=UzTϕf. Let Aut(𝔻) be the Lie group of all automorphisms (biholomorphic mappings) of 𝔻, and G0 the isotropy subgroup at 0; that is, G0={Ψ∈Aut(𝔻):Ψ(0)=0}.


## 2. Compact Operators Whose Real and Imaginary  Parts Are Positive

Zhu [1] had shown that if ϕ is a nonnegative function on 𝔻, 1≤p≤∞, then Tϕ is in the Schatten class Sp if and only if ϕ̃(z) is in Lp(𝔻,dλ). The following is an easy consequence of it.Proposition 2.1.
Suppose thatTϕ∈ℒ(La2(𝔻)) is such that Tϕ=Tϕ1+iTϕ2, where ϕ1≥0 and ϕ2≥0. The Toeplitz operator Tϕ∈Sp, 1≤p≤∞ if and only if ϕ̃(z)∈Lp(𝔻,dλ). In this case, ∥Tϕ∥p2≤∥Tϕ1∥p2+∥Tϕ2∥p2 if 2≤p<∞ and ∥Tϕ∥p2≥∥Tϕ1∥p2+∥Tϕ2∥p2 if 1≤p≤2.Proof.
Suppose2≤p≤∞ and Tϕ∈Sp. Notice that Tϕ1=(Tϕ+Tϕ*)/2 and Tϕ2=(Tϕ-Tϕ*)/2i and since Sp is a Banach space and is closed under adjoints, hence Tϕ1 and Tϕ2 belong to Sp. From [1], it follows that ϕ1̃,ϕ2̃∈Lp(𝔻,dλ). Hence ϕ̃=ϕ1̃+iϕ2̃∈Lp(𝔻,dλ). Now suppose ϕ̃∈Lp(𝔻,dλ), 2≤p≤∞. This implies ϕ1̃,ϕ2̃∈Lp(𝔻,dλ). From [1], it follows that Tϕ1,Tϕ2∈Sp. Hence Tϕ∈Sp as Sp is a vector space and ∥Tϕ∥p2≤∥Tϕ1∥p2+∥Tϕ2∥p2 (see [6]). Now let 1≤p<2 and assume Tϕ∈Sp. This implies Tϕ1,Tϕ2∈Sp and therefore by [1], ϕ1̃,ϕ2̃∈Lp(𝔻,dλ). Hence ϕ̃∈Lp(𝔻,dλ) and by [6], ∥Tϕ∥p2≥∥Tϕ1∥p2+∥Tϕ2∥p2. Now suppose ϕ̃∈Lp(𝔻,dλ), 1≤p<2. Then ϕ1̃,ϕ2̃∈Lp(𝔻,dλ) and hence Tϕ1,Tϕ2∈Sp. Since Sp is a vector space, Tϕ∈Sp.Example 2.2.
Letϕ(z)=(1-|z|)r, r>0. Then Tϕ∈S2 only if r>1/2, and Tϕ∈S1 only if r>1. This can be seen as follows. The matrix of Tϕ with respect to the standard orthonormal basis {en(z)}n≥0={n+1zn}n≥0 of La2(𝔻) is diagonal, tkk=〈Tϕek,ek〉=(2k+2)!/((r+1)(r+2)⋯(r+2k+2)) and ∑k=0∞|tkk|2=∞ for r≤1/2. Similarly ∑k=0∞tkk<∞ if r>1. If r≤1, ∑k=0∞tkk=∞.Proposition 2.3.
Letϕ∈L∞(𝔻) and suppose that ϕ is not the zero function. If Tϕ is compact then RangeTϕ is not closed.Proof.
SinceTϕ is compact, hence RangeTϕ contains no closed infinite-dimensional subspace of La2(𝔻). If now RangeTϕ is closed then RangeTϕ is finite dimensional. That is, Tϕ is of finite rank. This implies by [7] that ϕ≡0. This is a contradiction as ϕ is not the zero function.Recall the following.Suppose thatA is a positive operator on a Hilbert space H and x is unit vector in H, then (i) 〈Apx,x〉≥〈Ax,x〉p for all p≥1; (ii) 〈Apx,x〉≤〈Ax,x〉p for all 0<p≤1. For proof see [1].Suppose thatφ:𝔻→𝔻 is analytic. Define the composition operator Cφ from La2(𝔻) into itself as Cφf=f∘φ. It is shown in [1] that Cφ is a bounded linear operator on La2(𝔻) and ∥Cφ∥≤(1+|φ(0)|)/(1-|φ(0)|). Given a∈𝔻 and f any measurable function on 𝔻, we define the function Caf by Caf(z)=f(ϕa(z)), where ϕa∈Aut(𝔻). The map Ca is a composition operator on La2(𝔻).Proposition 2.4.
Ifϕ∈L∞(𝔻) then Tϕ is compact if and only if Tϕ∘ϕz is compact.Proof.
This follows from the fact thatTϕ∘ϕz=UzTϕUz and as Uz2=I.Proposition 2.5.
Forϕ∈L∞(𝔻), MO(ϕ)2(z)=|ϕ|2̃(z)-|ϕ̃(z)|2≤∥Hϕkz∥2+∥Hϕ¯kz∥2.Proof.
Observe that(2.1)∥Hϕkz∥=∥(I-P)(ϕkz)∥=∥(I-P)Uz(ϕ∘ϕz)∥=∥Uz(I-P)(ϕ∘ϕz)∥=∥(I-P)(ϕ∘ϕz)∥=∥ϕ∘ϕz-P(ϕ∘ϕz)∥.
Similarly, we have ∥Hϕ¯kz∥=∥ϕ¯∘ϕz-P(ϕ¯∘ϕz)∥=∥ϕ∘ϕz-P(ϕ¯∘ϕz)¯∥. Since ϕ̃(z)=P(ϕ∘ϕz)(0) and Pg¯(z)=g¯(0) for any g∈La2 and all z∈𝔻, we have
(2.2)MO(ϕ)2(z)=|ϕ|2̃(z)-|ϕ̃(z)|2=∥ϕ∘ϕz-P(ϕ∘ϕz)(0)∥2=∥ϕ∘ϕz-P(ϕ∘ϕz)∥2+∥P(ϕ∘ϕz)-P(ϕ∘ϕz)(0)∥2=∥Hϕkz∥2+∥P(ϕ∘ϕz)-P(ϕ¯∘ϕz)(0)¯∥2=∥Hϕkz∥2+∥P(ϕ∘ϕz-P(ϕ¯∘ϕz)¯)∥2≤∥Hϕkz∥2+∥ϕ∘ϕz-P(ϕ¯∘ϕz)¯∥2=∥Hϕkz∥2+∥Hϕ¯kz∥2.Leth>1. The generalized Kantorvich constant K(p) is defined by
(2.3)K(p)=hp-h(p-1)(h-1)(p-1php-1hp-h)p
for any real number p and it is known that K(p)∈(0,1] for p∈[0,1]. We state below the known results on the generalized Kantorvich constant K(p). Let A be strictly positive operator satisfying MI≥A≥mI>0, where M>m>0. Put h=M/m>1. Then the following [8] inequalities (2.4) and (2.5) hold for every unit vector x and are equivalent:
(2.4)K(p)〈Ax,x〉p≥〈Apx,x〉≥〈Ax,x〉pforanyp>1oranyp<0;(2.5)〈Ax,x〉p≥〈Apx,x〉≥K(p)〈Ax,x〉pforanyp∈(0,1].The Kantorvich constantK(p) is symmetric with respect to p=1/2 and K(p) is an increasing function of p for p≥1/2, K(p) is a decreasing function of p for p≤1/2, and K(0)=K(1)=1. Further, K(p)≥1 for p≥1 or p≤0, and 1≥K(p)≥2h1/4/(h1/2+1) for p∈[0,1].Proposition 2.6.
LetTϕ be strictly positive satisfying MI≥Tϕ≥mI>0, where M>m>0. The following hold.(i)
If0<p<∞ and Tϕ∈Sp then ϕ̃∈Lp(𝔻,dλ).(ii)
If0<p≤1, ϕ̃∈Lp(𝔻,dλ) then Tϕ∈Sp.(iii)
Letp∈[1,∞) be such that K(p)<∞. If ϕ̃∈Lp(𝔻,dλ) then Tϕ∈Sp.Proof.
Supposep>1 and Tϕ∈Sp. Then
(2.6)∫𝔻〈Tϕpkz,kz〉dλ(z)=∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞.
Hence by (2.4), ∫𝔻〈Tϕkz,kz〉pdλ(z)<∞. That is, ϕ̃∈Lp(𝔻,dλ). Suppose 0<p≤1 and Tϕ∈Sp. Then ∫𝔻〈Tϕpkz,kz〉dλ(z)=∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞. Hence from (2.5), it follows that K(p)∫𝔻〈Tϕkz,kz〉pdλ(z)<∞. Since K(p)∈(0,1] for p∈[0,1], hence ϕ̃∈Lp(𝔻,dλ).
Now assumeϕ̃∈Lp(𝔻,dλ). Then if 0<p≤1 then by (2.5), we have ∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞ and hence Tϕ∈Sp. If 1≤p<∞, then by (2.4) and (2.5), if K(p)<∞ and ϕ̃∈Lp(𝔻,dλ) then ∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞ and Tϕ∈Sp.


## 3. Schatten Class Operators

In this section, we will obtain conditions to describe Schatten class Toeplitz operators on Bergman spaceLa2(𝔻). The results of this paper hold for Bergman spaces La2(Ω), where Ω is any bounded symmetric domain in ℂ. For simplicity, we consider only the case of the open unit disk 𝔻 in ℂ.LetBT={f∈L1:∥f∥BT=supz∈𝔻|f|̃(z)<∞}. The space L∞ is properly contained in BT (see [9]) and if ϕ∈BT then Tϕ is bounded on La2 and there is a constant C such that ∥Tϕ∥≤C∥ϕ∥BT.Theorem 3.1.
Suppose1≤p<∞ and dλ(z)=dA(z)/(1-|z|2)2. Then the following hold. (1) If Tϕ∈Sp, then ϕ̃∈Lp(𝔻,dλ). (2) If ϕ∈Lp(𝔻,dλ) then ϕ̃∈Lp(𝔻,dλ) and Tϕ∈Sp.Proof.
SupposeTϕ∈Sp. Then
(3.1)∫𝔻〈|Tϕ|pkw,kw〉dλ(w)<∞.
That is, ∫𝔻〈(Tϕ*Tϕ)p/2kw,kw〉dλ(w)<∞. If 2≤p<∞, then
(3.2)∫𝔻〈Tϕ*Tϕkw,kw〉p/2dλ(w)≤∫𝔻〈(Tϕ*Tϕ)p/2kw,kw〉dλ(w)<∞.
This implies
(3.3)∫𝔻∥P(ϕ∘ϕw)∥pdλ(w)=∫𝔻∥P(Uw(ϕkw))∥pdλ(w)=∫𝔻∥UwTϕkw∥pdλ(w)=∫𝔻∥Tϕkw∥pdλ(w)=∫𝔻〈Tϕ*Tϕkw,kw〉p/2dλ(w)<∞.
Now |P(ϕ∘ϕw)(0)|=|〈P(ϕ∘ϕw),1〉|=|〈Uw(Tϕkw),1〉|=|〈Tϕkw,Uw1〉|=|〈Tϕkw,kw〉|≤∥Tϕkw∥=∥P(ϕ∘ϕw)∥. Thus ∫𝔻|P(ϕ∘ϕw)(0)|pdλ(w)<∞. That is, ∫𝔻|ϕ̃(w)|pdλ(w)<∞ and ϕ̃∈Lp(𝔻,dλ). Suppose 1≤p<2. Then by Heinz inequality [10], it follows that
(3.4)∞>∫𝔻〈|Tϕ|pkw,kw〉dλ(w)=∫𝔻〈|Tϕ|2·p/2kw,kw〉dλ(w)≥∫𝔻|〈Tϕkw,kw〉|2〈|Tϕ*|2(1-p/2)kw,kw〉dλ(w)=∫𝔻|ϕ̃(w)|2∥P(ϕ¯∘ϕw)∥2-pdλ(w)=∫𝔻|ϕ̃(w)|2∥P(ϕ¯∘ϕw)∥p-2dλ(w)≥∫𝔻|ϕ̃(w)|2∥P(ϕ¯∘ϕw)∥2∥P(ϕ¯∘ϕw)∥pdλ(w)≥∫𝔻|ϕ̃(w)|2C2∥ϕ∥BT2|P(ϕ∘ϕw)(0)|pdλ(w)=∫𝔻|ϕ̃(w)|2C2∥ϕ∥BT2|ϕ̃(w)|pdλ(w),
since
(3.5)〈|Tϕ*|2-pkw,kw〉=〈|Tϕ*|2·(2-p)/2kw,kw〉≤〈|Tϕ*|2kw,kw〉(2-p)/2=〈TϕTϕ*kw,kw〉(2-p)/2=∥Tϕ*kw∥2-p=∥P(ϕ̅∘ϕw)∥2-p.
Hence(3.6)∫𝔻|ϕ̃(w)|p+2dλ(w)<∞,
and therefore ∫𝔻|ϕ̃(w)|pdλ(w)<∞. Thus ϕ̃∈Lp(𝔻,dλ).
Now supposeϕ∈L1(𝔻,dλ). Then the change of the order of integration
(3.7)∫𝔻|ϕ̃(w)|dλ(w)=∫𝔻|ϕ̃(w)|dA(w)(1-|w|2)2≤∫𝔻(∫𝔻|ϕ(z)|(1-|w|2)2|1-w̅z|4dA(z))dA(w)(1-|w|2)2=∫𝔻|ϕ(z)|∫𝔻dA(w)|1-w̅z|4dA(z)=∫𝔻|ϕ(z)|〈kz,kz〉dA(z)=∫𝔻|ϕ(z)|dA(z)(1-|z|2)2
is justified by the positivity of the integrand. Hence ϕ̃∈L1(𝔻,dλ).  Similarly if ϕ∈L∞(𝔻) then ϕ̃∈L∞(𝔻) as |ϕ̃(w)|=|〈ϕkw,kw〉|≤∥ϕkw∥2∥kw∥2≤∥ϕ∥∞∥kw∥22=∥ϕ∥∞. By Marcinkiewicz interpolation theorem it follows that if ϕ∈Lp(𝔻,dλ) then ϕ̃∈Lp(𝔻,dλ) for 1≤p≤∞. Now suppose ϕ∈Lp(𝔻,dλ), 1≤p≤∞. We will prove Tϕ∈Sp. The case p=+∞ is trivial. By interpolation we need only to prove the result for p=1. Suppose ϕ∈L1(𝔻,dλ) and {en}={n+1zn}n=0∞ is the standard orthonormal basis for La2(𝔻). Now 〈Tϕen,en〉=∫𝔻|en(z)|2ϕ(z)dA(z) and
(3.8)∑n=0∞|〈Tϕen,en〉|≤∫𝔻∑n=0∞|en(z)|2|ϕ(z)|dA(z)≤∫𝔻K(z,z)|ϕ(z)|dA(z)=∫𝔻|ϕ(z)|dλ(z).
Thus Tϕ∈S1 and ∥Tϕ∥S1≤∫𝔻|ϕ(z)|dλ(z).It is not so difficult to verify the conditions in Theorem3.1.Example 3.2.
LetΦ(z)=(1-|z|2)log(1-|z|2). Then Φ∈L2(𝔻,dλ). This can be verified as follows:
(3.9)∫𝔻(1-|z|2)2|log(1-|z|2)|2dλ(z)=∫𝔻(1-|z|2)2|log(1-|z|2)|2dA(z)|(1-|z|2)|2=∫𝔻|log(1-|z|2)|2dA(z)=∫01|log(1-r2)|22rdr=∫01|log(1-t)|2dt=∫01|logt|2dt,
and changing the variable to y=-logt, this reduces to ∫0∞y2e-ydy=Γ(3)=2<∞. Thus TΦ∈S2.Example 3.3.
Letg(z)=ln|z|2. Then
(3.10)∫𝔻|g(z)|2dλ(z)=∫01(lnt1-t)2dt=∫01∑n=0∞∑m=0∞tm+nln2tdt=∑n=0∞∑m=0∞2(m+n+1)3=∑k=0∞2(k+1)2=π23<∞.
Thus g∈L2(𝔻,dλ). Direct computation reveals that g̃(x)=|x|2-1 and g̃∈L2(𝔻,dλ).Example 3.4.
Letϕ(z)=1-|z|. Then by Example 2.2, Tϕ∈S2 and
(3.11)∫𝔻|ϕ(z)|2dλ(z)=∫𝔻(1-|z|)2dA(z)(1-|z|2)2=∫𝔻dA(z)(1+|z|)2≤∫𝔻dA(z)=1.
Thus ϕ∈L2(𝔻,dλ) and hence ϕ̃∈L2(𝔻,dλ).
It may be noted that the spaceLp(𝔻,dλ), 1≤p<∞, contains no nonzero harmonic functions and even no nonzero constants. To see this, for example, for L2(𝔻,dλ), let
(3.12)M(r)=12π∫02π|f(reit)|2dt.
This is a nonnegative and nondecreasing function of r and ∥f∥L2(𝔻,dλ)2=∫01M(r)(2r/(1-r2)2)dr<∞. So M(r) must tend to 0 as r→1. Thus M(r)≡0, and therefore f≡0.But although there is no nonzero harmonic functions inL2(𝔻,dλ), there are plenty of subharmonic functions. Consider the function f(z)=ln|z|2. We have verified in Example 3.3 that f∈L2(𝔻,dλ). Suppose that f is real-valued subharmonic and f∈L2(𝔻,dλ). The subharmonicity of f implies that f̃(w)=∫𝔻f(ϕw(z))dA(w)≥f(ϕw(0))=f(w). Hence f̃≥f. Let Δh:=(1-|z|2)2(∂2/∂z∂z̅). It can be verified that Δh(f∘ϕa)=(Δhf)∘ϕa and note that Δhf̃=Δhf̃≥0 since Δhf≥0. In other words, f̃ is also subharmonic. Proceeding by induction, if we define Bg=g̃ on L2(𝔻,dλ) then we obtain Bnf is subharmonic for all n∈ℕ and {Bnf}n∈ℕ is a nondecreasing sequence of functions.Corollary 3.5.
Ifϕ∈h∞(𝔻), the space of bounded harmonic functions on 𝔻 and 1≤p<∞, then Tϕ∈Sp if and only if ϕ≡0.Proof.
The proof of the corollary follows from the above discussion and the fact [1] that ϕ∈h∞(𝔻) if and only if ϕ̃=ϕ.Corollary 3.6.
Ifϕ is a real-valued bounded subharmonic function on 𝔻, 1≤p<∞, then Tϕ∈Sp if and only if ϕ̃∈Lp(𝔻,dλ).Proof.
By Theorem3.1, if Tϕ∈Sp then ϕ̃∈Lp(𝔻,dλ). Conversely if ϕ is real-valued, subharmonic, bounded on 𝔻 and ϕ̃∈Lp(𝔻,dλ) then ϕ̃ is also subharmonic and the subharmonicity of ϕ implies that ϕ̃(w)=∫𝔻ϕ(ϕw(z))dA(z)≥ϕ(ϕw(0))=ϕ(w). Hence ∫𝔻|ϕ̃(w)|pdλ(w)=∫𝔻|ϕ(w)|pdλ(w) as ϕ∈Lp(𝔻,dλ) implies ϕ̃∈Lp(𝔻,dλ) and the result follows from Theorem 3.1.Corollary3.7 follows immediately from Corollary 3.5. We present a proof of Corollary 3.7 to show a different method of approach.Corollary 3.7.
Ifϕ∈H∞(𝔻)∪H∞(𝔻)¯ then the Toeplitz operator Tϕ∈Sp,1≤p<∞ if and only if ϕ≡0.Proof.
Notice that ifϕ∈H∞(𝔻)∪H∞(𝔻)¯ then the following two possibilities hold.(*)
For allz∈𝔻, either P(ϕ∘ϕz)≠P(ϕ∘ϕz)(0) or P(ϕ̅∘ϕz)≠P(ϕ̅∘ϕz)(0).(**)
There existsz∈D such that P(ϕ∘ϕz)=P(ϕ∘ϕz)(0) and P(ϕ̅∘ϕz)=P(ϕ̅∘ϕz)(0).
Suppose that (**) holds and let z∈𝔻 be such that P(ϕ∘ϕz)=P(ϕ∘ϕz)(0) and P(ϕ̅∘ϕz)=P(ϕ̅∘ϕz)(0). Then this implies Tϕkz=czkz and Tϕ̅kz=c̅zkz. Hence Tϕ has an eigenvalue. From [1], it follows that ϕ is a constant and Tϕ is not compact unless ϕ≡0.
Now suppose that (*) holds, 1≤p<∞. Then for all z∈𝔻, ∥Hϕkz∥2+∥Hϕ¯kz∥2=∥ϕ∘ϕz-P(ϕ∘ϕz)∥2+∥ϕ¯∘ϕz-P(ϕ¯∘ϕz)∥2<∥ϕ∘ϕz-P(ϕ∘ϕz)(0)∥2+∥ϕ¯∘ϕz-P(ϕ¯∘ϕz)(0)∥2=2(|ϕ|2̃(z)-|ϕ̃(z)|2). Let cz=(∥Hϕkz∥2+∥Hϕ¯kz∥2)/(|ϕ|2̃(z)-|ϕ̃(z)|2). Note that 1≤cz<2 for all z∈𝔻. It follows from Proposition 2.5 and from the previous discussion.
NoticeTϕ compact implies that Tϕ*Tϕ=T|ϕ|2-Hϕ*Hϕ is compact. Thus 0≤〈Tϕ*Tϕkz,kz〉=|ϕ|2̃(z)-∥Hϕkz∥2→0 as |z|→1-. Similarly since TϕTϕ* is compact, 0≤|ϕ|2̃(z)-∥Hϕ¯kz∥2→0 as |z|→1-.
Thus0≤2|ϕ|2̃(z)-(∥Hϕkz∥2+∥Hϕ¯kz∥2)→0 as |z|→1-. Hence it follows that 0≤2|ϕ|2̃(z)-cz(|ϕ|2̃(z)-|ϕ̃(z)|2)=2|ϕ|2̃(z)-(∥Hϕkz∥2+∥Hϕ¯kz∥2)→0 as |z|→1-. So 0≤(2-cz)|ϕ|2̃(z)+cz|ϕ̃(z)|2→0 as |z|→1-, where 1≤cz<2. Since Tϕ is compact, |ϕ̃(z)|→0 as |z|→1-. Thus |ϕ|2̃(z)→0 as |z|→1-. Similarly since Tϕ∘ϕz is compact we can show that |ϕ∘ϕz|2̃(z)→0 as |z|→1-. Thus
(3.13)∫𝔻|(ϕ∘ϕz)(w)|2|kz(w)|2dA(w)→0
as |z|→1-. Hence ∫𝔻|ϕ(w)|2dA(w)=0 as Uzkz=kz(ϕz(w))kz(w)=1. It follows therefore that ϕ(w)=0 almost everywhere and hence ϕ≡0. Therefore, Tϕ≡0.LetT̃(z)=〈Tkz,kz〉 for T∈ℒ(La2(𝔻)). It follows from Heinz inequality [10] that
(3.14)|〈Tϕkz,kz〉|2≤〈|Tϕ|2αkz,kz〉〈|Tϕ*|2(1-α)kz,kz〉
for all z∈𝔻 and 0≤α≤1. That is, |ϕ̃(z)|2≤|Tϕ|2α̃(z)|Tϕ*|2(1-α)̃(z) for all z∈𝔻. Hence if |Tϕ|2α̃∈Lp(𝔻,dλ) and |Tϕ*|2(1-α)̃∈Lq(𝔻,dλ), 1/p+1/q=1, it follows from Holders inequality that ϕ̃∈L2(𝔻,dλ). From Theorem 3.1, it follows that Tϕ̃∈S2.We sayT majorizes S∈ℒ(La2(𝔻)) if ∥Sf∥≤M∥Tf∥ for all f∈La2(𝔻).Corollary 3.8.
Ifϕ≥0, ϕ̃∈Lp(𝔻,dλ), 1≤p≤2, and Tϕ majorizes Tψ then Tψ∈Sp.Proof.
SinceTϕ majorizes Tψ, it follows that ∥Tψf∥≤M∥Tϕf∥ for some M>0 and for all f∈La2. Hence 〈Tψ*Tψf,f〉≤M2〈Tϕ*Tϕf,f〉 for all f∈La2. That is, Tψ*Tψ≤M2Tϕ*Tϕ. Since 1≤p≤2, we obtain from [10, 11] that (Tψ*Tψ)p/2≤Mp(Tϕ*Tϕ)p/2. That is, |Tψ|p≤Mp|Tϕ|p for 1≤p≤2. Now if ϕ̃∈Lp(𝔻,dλ), ϕ≥0, then by [1], Tϕ∈Sp and ∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞. Hence ∫𝔻〈|Tψ|pkz,kz〉dλ(z)<∞ and Tψ∈Sp.Corollary 3.9.
Letϕ,ψ∈L∞(𝔻). If RangeTψ̅⊆RangeTϕ̅ and ϕ∈Lp(𝔻,dλ), 1≤p≤2 then Tψ∈Sp and ψ̃∈Lp(𝔻,dλ).Proof.
RangeTψ̅⊆RangeTϕ̅ implies that Tϕ majorizes Tψ. Hence ∥Tψf∥≤M∥Tϕf∥ for some M>0 and for all f∈La2. Hence 〈Tψ*Tψf,f〉≤M2〈Tϕ*Tϕf,f〉 for all f∈La2. That is, Tψ*Tψ≤M2Tϕ*Tϕ. Since 1≤p≤2, we obtain from [10, 11] that (Tψ*Tψ)p/2≤Mp(Tϕ*Tϕ)p/2. That is, |Tψ|p≤Mp|Tϕ|p for 1≤p≤2. Now if ϕ∈Lp(𝔻,dλ) then Tϕ∈Sp and ∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞. Hence ∫𝔻〈|Tψ|pkz,kz〉dλ(z)<∞. Thus Tψ∈Sp and ψ̃∈Lp(𝔻,dλ).Definition 3.10.
A functionG∈La2(D)(G∈H2) is called an inner function [12] in La2(D) (resp., H2) if |G|2-1 is orthogonal to H∞. If for some inner function G∈La2(D) the following conditions hold then we call it a finite zero divisor in La2(D). (i) G vanishes on a={aj}j=1N, a finite sequence of points in D. (ii) ∥G∥L2=1. (iii) G is equal to a constant plus a linear combination of the Bergman kernel functions K(z,a1),K(z,a2),…,K(z,an) and certain of their derivatives. (iv) |G|2-1 is orthogonal to Lh1, the class of harmonic functions in L1 of the disc.Corollary 3.11.
Supposeϕ∈L∞(𝔻) and kerTϕ⊆GLa2, where G is a finite inner divisor in La2(𝔻). If ϕ∈Lp(𝔻,dλ),1≤p≤2, then Sψ∈Sp, where Sψ is the little Hankel operator on La2(𝔻) such that kerSψ=GLa2.Proof.
Letb=(bj)j=1N be a finite set of points in 𝔻 that are the zeroes of the finite inner divisor G counting multiplicities and ℐ=I(b)={f∈La2(D):f=0onb}. Then G is the solution of the extremal problem
(3.15)sup{Ref(n)(0):f∈ℐ,∥f∥L2≤1},
where n is the number of times zero appears in the sequence b (i.e., the functions in ℐ have a common zero of order n at the origin). It is shown in [12] that for
(3.16)ψ̅=∑j=1N∑ν=0mj-1cjν∂ν∂b̅jνKbj(z),
where cjν≠0 for all j,ν and mj is the number of times bj appear in b, kerSψ=GLa2(D). Now kerTϕ⊆GLa2=kerSψ implies [13] the operator Tϕ that majorizes Sψ. Since ϕ∈Lp(𝔻,dλ), 1≤p≤2, Tϕ∈Sp. By similar arguments as in Corollary 3.9 one can show that Sψ∈Sp.For anya∈𝔻, let γa be the unique geodesic such that γa(0)=0, γa(1)=a. Then there exists a unique ϕa∈Aut(𝔻) such that ϕa∘ϕa(z)≡z and γa(1/2) is an isolated fixed point of ϕa. Further ϕa is the geodesic symmetry at γa(1/2). We denote by ma the geodesic midpoint γa(1/2) of 0 and a. It can easily be verified that each ϕa has ma as a unique fixed point and for a∈𝔻,  ma=((1-1-|a|2)/|a|2)a.Givenλ∈𝔻 and a measurable function f on 𝔻, we have f∘ϕλ=f if and only if there exists an even function g on 𝔻 such that f=g∘ϕmλ; f∘ϕλ=-f if and only if there exists an odd function g on 𝔻 such that f=g∘ϕmλ. For proof of this fact see [14].Corollary 3.12.
Ifϕ=θ∘ϕmz for some z∈𝔻, θ∈L∞(𝔻) and where θ is an even function and Tϕ∈Sp, 1≤p<∞, then
(3.17)∥Tψ∘ϕz-ψ+ϕ∥Sp≥∥Tϕ∥Sp
for all Tψ∈Sp.Proof.
Sinceϕ=θ∘ϕmz where θ is an even function, by [14], we have ϕ∘ϕz=ϕ. Hence UzTϕUz=Tϕ∘ϕz=Tϕ. As Uz is unitary, from [15], it follows that
(3.18)∥UzTψUz-Tψ+Tϕ∥Sp≥∥Tϕ∥Sp
for all Tψ∈Sp. Hence
(3.19)∥Tψ∘ϕz-ψ+ϕ∥Sp≥∥Tϕ∥Sp
for all Tψ∈Sp.Theorem 3.13.
Suppose there exist constantsC,M>0 such that C>M and ∥P(ϕ∘ϕz)-C∥≤M for all z∈𝔻, ϕ̃∈Lp(𝔻,dλ) and p∈[1,∞) is such that the Kantorvich constant K(p)∈(0,∞). If Tϕ is bounded below then Tϕ∈Sp.Proof.
If there existC>M>0 such that for all z∈𝔻, we have ∥P(ϕ∘ϕz)-C∥≤M then ∥Tϕkz-Ckz∥=∥UzP(ϕ∘ϕz)-CUz1∥=∥P(ϕ∘ϕz)-C∥≤M for all z∈𝔻. Let Γ=C+M and γ=C-M. Then 0<γ<Γ and C=(Γ+γ)/2 and M=(Γ-γ)/2. Thus for all z∈𝔻, ∥Tϕkz-((Γ+γ)/2)kz∥≤(1/2)|Γ-γ|∥kz∥. By [16], it follows that there exists L>0 such that |ϕ̃(z)|=|〈Tϕkz,kz〉|≥L∥Tϕkz∥ for all z∈𝔻. Notice that the reproducing kernels {kz:z∈𝔻} span La2(𝔻) and Tϕ is bounded below. Hence there exist r>0, s>0 such that s≤Tϕ*Tϕ≤r. Now if ϕ̃∈Lp(𝔻,dλ) and p∈[1,∞) is such that the Kantorvich constant K(p)∈(0,∞) then ∫𝔻∥Tϕkz∥pdλ(z)<∞. From inequalities (2.4) and (2.5), it follows that ∫𝔻〈|Tϕ|pkz,kz〉dλ(z)<∞. Hence Tϕ∈Sp.LetL2(𝕋) be the usual Lebesgue space of functions on the unit circle 𝕋, and let the Hardy space H2 be the class of all L2(𝕋) functions whose negative Fourier coefficients are zero. Let L∞(𝕋) be the algebra of essentially bounded complex valued functions on the unit circle 𝕋 and H∞(𝕋) be the subalgebra of L∞(𝕋) consisting of functions whose negative Fourier coefficients are zero. Let ℛn denote the rational functions with at most n poles (counting multiplicities) all of which are in the interior of 𝕋, and let C(𝕋) denote the algebra of continuous functions on 𝕋. Notice that the following inclusion relations hold:
(3.20)H∞(𝕋)⊂H∞(𝕋)+ℛ1⊂H∞(𝕋)+ℛ2⊂⋯H∞(𝕋)+C(𝕋).
For ϕ∈L∞(𝕋), let Bϕ:H2→H2 be the Toeplitz operator on H2 with symbol ϕ, and let Γϕ:H2→H2 be the Hankel operator with symbol ϕ. Nehari [17], Adamjan et al. [5], and Hartman [18], respectively, proved the following results:(i)
∥Γϕ∥=d(ϕ,H∞(𝕋));(ii)
sk(Γϕ)=d(ϕ,H∞(𝕋)+ℛn);(iii)
∥Γϕ∥e=d(ϕ,H∞(𝕋)+C(𝕋)),where∥Γϕ∥e denotes the essential norm of Γϕ. Feintuch in [19, 20] obtained the following operator theoretic analogues of these distance formulae. Given T∈ℒ(H2(𝕋)), define a sequence {Γn(𝕋)} of operators on H2(𝕋) by Γn(𝕋)=JnTSn+1, n=1,2,…, where for z∈𝕋,
(3.21)Jnzi={zn-i,if0≤i≤n,0,ifi>n,
and S is the unilateral shift on H2(𝕋). Let 𝒯̂ denote the family of operators T for which {Γn(𝕋)} converges strongly and
(3.22)ℒ0={T∈𝒯̂:Γn(T)convergesstronglyto0};ℒk=ℒ0+{Bϕ:ϕ∈H∞(𝕋)+ℛk};ℒ∞=ℒ0+{Bϕ:ϕ∈H∞(𝕋)+C(𝕋)}.Then (i)d(T,ℒ0)=∥Γ(𝕋)∥; (ii) d(T,ℒk)=sk(Γ(𝕋)); (iii) d(T,ℒ∞)=∥Γ(𝕋)∥e. In particular when T=Bϕ, ϕ∈L∞(𝕋) we have(i)
d(Bϕ,ℒ0)=∥Γϕ∥=d(ϕ,H∞(𝕋));(ii)
d(Bϕ,ℒk)=sk(Γϕ)=d(ϕ,H∞(𝕋)+ℛk);(iii)
d(Bϕ,ℒ∞)=∥Γϕ∥e=d(ϕ,H∞(𝕋)+C(𝕋)).Yamada [21] also obtained distance formulas involving the norm of Toeplitz operators on the Hardy space.It is well known that [1] there are no compact Toeplitz operators on the Hardy space other than the zero operator. In the Bergman space setting, however, there are lots of nontrivial compact Toeplitz operators belonging to different Schatten classes. In view of this it is of interest to know whether such distance formulae is possible for Toeplitz and Hankel operators defined on the Bergman space and the characterization of Schatten class Toeplitz operators is also important in this context. These results play an important role in approximation theory.



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*Source: 921324-2009-11-11.xml*