Schatten Class Toeplitz Operators on the Bergman Space

Abstract

We have shown that if the Toeplitz operator 𝑇_𝜙 on the Bergman space 𝐿²_𝑎(𝔻) belongs to the Schatten class 𝑆_𝑝,1≤𝑝<∞, then ^0𝑥0𝜙^02𝑑𝑐∈𝐿^𝑝(𝔻,𝑑𝜆), where ^0𝑥0𝜙^02𝑑𝑐 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∈𝐿^𝑝(𝔻,𝑑𝜆) then ^0𝑥0𝜙^02𝑑𝑐∈𝐿^𝑝(𝔻,𝑑𝜆) and 𝑇_𝜙∈𝑆_𝑝. For certain subclasses of 𝐿^∞(𝔻), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.

1. Introduction

Let 𝔻={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 ⟨𝑓,𝑔⟩=𝑓(𝑧)𝑔(𝑧)𝑑𝐴(𝑧). (1.1) Let 𝐿^∞(𝔻,𝑑𝐴) denote the Banach space of Lebesgue measurable functions 𝑓 on 𝔻 with ‖𝑓‖_∞=esssup{|𝑓(𝑧)|∶𝑧∈𝔻}<∞, (1.2) and let 𝐻^∞(𝔻) be the space of bounded analytic functions on 𝔻. Let 𝐿²_𝑎(𝔻) be the subspace of 𝐿²(𝔻,𝑑𝐴) consisting of analytic functions. The space 𝐿²_𝑎(𝔻) is called the Bergman space. Since point evaluation at 𝑧∈𝔻 is a bounded linear functional on the Hilbert space 𝐿²_𝑎(𝔻), the Riesz representation theorem implies that there exists a unique function 𝐾_𝑧 in 𝐿²_𝑎(𝔻) such that 𝑓(𝑧)=_𝔻𝑓(𝑤)𝐾_𝑧(𝑤)𝑑𝐴(𝑤) (1.3) for all 𝑓 in 𝐿²_𝑎(𝔻). Let 𝐾(𝑧,𝑤) be the function on 𝔻×𝔻 defined by 𝐾(𝑧,𝑤)=𝐾_𝑧(𝑤). (1.4) The function 𝐾(𝑧,𝑤) is thus the reproducing kernel for the Bergman space 𝐿²_𝑎(𝔻) and is called the Bergman kernel. It can be shown that the sequence of functions {𝑒_𝑛(𝑧)}={√𝑛+1𝑧^𝑛}_𝑛≥0 forms the standard orthonormal basis for 𝐿²_𝑎(𝔻) and 𝐾(𝑧,𝑤)=∑^∞_𝑛=1𝑒_𝑛(𝑧)𝑒_𝑛(𝑤). The Bergman kernel is independent of the choice of orthonormal basis and 𝐾(𝑧,𝑤)=1/(1−𝑧𝑤)².

Let 𝑘_𝑎(𝑧)=𝐾(𝑧,𝑎)/𝐾(𝑎,𝑎)=(1−|𝑎|²)/(1−𝑎𝑧)². These functions 𝑘_𝑎 are called the normalized reproducing kernels of 𝐿²_𝑎; it is clear that they are unit vectors in 𝐿²_𝑎. For any 𝑎∈𝔻, let 𝜙_𝑎 be the analytic mapping on 𝔻 defined by 𝜙_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧), 𝑧∈𝔻. An easy calculation shows that the derivative of 𝜙_𝑎 at 𝑧 is equal to −𝑘_𝑎(𝑧). It follows that the real Jacobian determinant of 𝜙_𝑎 at 𝑧 is 𝐽_𝜙𝑎(𝑧)=|𝑘_𝑎(𝑧)|²=(1−|𝑎|²)²|1−𝑎𝑧|⁴. (1.5) When |𝑧|→1, 𝑘_𝑧→0 weakly and the normalized reproducing kernels 𝑘_𝑧, 𝑧∈𝔻 span 𝐿²_𝑎(𝔻) [1]. Since 𝐿²_𝑎(𝔻) is a closed subspace of 𝐿²(𝔻,𝑑𝐴) (see [1]), there exists an orthogonal projection 𝑃 from 𝐿²(𝔻,𝑑𝐴) onto 𝐿²_𝑎(𝔻). For 𝜙∈𝐿^∞(𝔻), we define the Toeplitz operator 𝑇_𝜙 on 𝐿²_𝑎(𝔻) by 𝑇_𝜙𝑓=𝑃(𝜙𝑓), 𝑓∈𝐿²_𝑎(𝔻). For 𝜙∈𝐿^∞(𝔻), let 𝜙(𝑧)=⟨𝑇_𝜙𝑘_𝑧,𝑘_𝑧⟩. The function 𝜙(𝑧) is called the Berezin transform of 𝜙. Let 𝐻_𝜙 be the Hankel operator from 𝐿²_𝑎 into (𝐿²_𝑎)^⟂ defined by 𝐻_𝜙𝑓=(𝐼−𝑃)(𝜙𝑓). It is easy to check that 𝐻^∗_𝜙𝐻_𝜙=𝑇_|𝜙|²−𝑇_𝜙𝑇_𝜙 (see [1]). For 𝜓∈𝐿^∞(𝔻), define 𝑆_𝜓∶𝐿²_𝑎→𝐿²_𝑎 as 𝑆_𝜓𝑓=𝑃𝐽(𝜓𝑓), where 𝐽∶𝐿²→𝐿² is defined as 𝐽𝑓(𝑧)=𝑓(𝑧). The operator 𝑆_𝜓 is called the little Hankel operator on 𝐿²_𝑎(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 entries 𝑎_𝑖𝑗 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 ideal 𝒯 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 integer 𝑛, the 𝑛th singular value of 𝑇∈ℒ𝒞(𝐻) is defined by 𝑠_𝑛(𝑇)=inf{‖𝑇−𝐾‖∶𝐾∈ℒ(𝐻),rank𝐾≤𝑛}. (1.6) Here ‖⋅‖ is the operator norm. Clearly, 𝑠₀(𝑇)=‖𝑇‖ and 𝑠₀(𝑇)≥𝑠₁(𝑇)≥𝑠₂(𝑇)≥⋯≥0. (1.7)

Thus 𝑠_𝑛(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 𝑠_𝑛(𝑇) can be defined for any 𝑇∈ℒ(𝐻) but clearly, 𝑠_𝑛(𝑇)→0 if and only if 𝑇 is compact.

The Schatten Von Neumann class 𝑆_𝑝=𝑆_𝑝(𝐻), 0<𝑝<∞, consists of all operators 𝑇∈ℒ𝒞(𝐻) such that ‖𝑇‖_𝑆𝑝=(^∞_𝑛=0(𝑠_𝑛(𝑇))^𝑝)^1/𝑝<∞. (1.8)

If 1≤𝑝<∞, then ‖⋅‖_𝑆𝑝 is a norm, which makes 𝑆_𝑝 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∈𝑆_𝑝 and 𝐶, a constant), which makes 𝑆_𝑝 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 trace𝑇=^∞_𝑛=0⟨𝑇𝜀_𝑛,𝜀_𝑛⟩,𝑇∈𝑆₁, (1.9) where {𝜀_𝑛}_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 𝑆^∗_𝑝 can be identified with 𝑆_𝑞 with respect to the pairing ⟨𝑇,𝑅⟩=trace𝑇𝑅^∗, 𝑇∈𝑆_𝑝, 𝑅∈𝑆_𝑞. Here 𝑞=𝑝/(𝑝−1) is the dual exponent. With respect to the same pairing one can identify 𝑆^∗₁ 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 on 𝜙 that will ensure that the Toeplitz operator 𝑇_𝜙 belong to 𝑆_𝑝, 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 𝐿²_𝑎(𝔻) into itself.

For 𝑧 and 𝑤 in 𝔻, let 𝜙_𝑧(𝑤)=(𝑧−𝑤)/(1−𝑧𝑤). These are involutive Mobius transformations on 𝐷. In fact

(1)𝜙_𝑧∘𝜙_𝑧(𝑤)≡𝑤;(2)𝜙_𝑧(0)=𝑧, 𝜙_𝑧(𝑧)=0;(3)𝜙_𝑧 has a unique fixed point in 𝔻.

Given 𝑧∈𝔻 and 𝑓 any measurable function on 𝔻, we define a function 𝑈_𝑧𝑓(𝑤)=𝑘_𝑧(𝑤)𝑓(𝜙_𝑧(𝑤)). Since |𝑘_𝑧|² is the real Jacobian determinant of the mapping 𝜙_𝑧 (see [1]), 𝑈_𝑧 is easily seen to be a unitary operator on 𝐿²(𝔻,𝑑𝐴) and 𝐿²_𝑎(𝔻). It is also easy to check that 𝑈^∗_𝑧=𝑈_𝑧, thus 𝑈_𝑧 is a self-adjoint unitary operator. If 𝜙∈𝐿^∞(𝔻,𝑑𝐴) and 𝑧∈𝔻 then 𝑈_𝑧𝑇_𝜙=𝑇_{𝜙∘𝜙𝑧}𝑈_𝑧. This is because 𝑃𝑈_𝑧=𝑈_𝑧𝑃 and for 𝑓∈𝐿²_𝑎, 𝑇_{𝜙∘𝜙𝑧}𝑈_𝑧𝑓=𝑇_{𝜙∘𝜙𝑧}((𝑓∘𝜙_𝑧)𝑘_𝑧)=𝑃((𝜙∘𝜙_𝑧)(𝑓∘𝜙_𝑧)𝑘_𝑧)=𝑃(𝑈_𝑧(𝜙𝑓))=𝑈_𝑧𝑃(𝜙𝑓)=𝑈_𝑧𝑇_𝜙𝑓. 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 𝑇_𝜙 is in the Schatten class 𝑆_𝑝 if and only if 𝜙(𝑧) is in 𝐿^𝑝(𝔻,𝑑𝜆). The following is an easy consequence of it.

Proposition 2.1. Suppose that 𝑇_𝜙∈ℒ(𝐿²_𝑎(𝔻)) is such that 𝑇_𝜙=𝑇_𝜙1+𝑖𝑇_𝜙2, where 𝜙₁≥0 and 𝜙₂≥0. The Toeplitz operator 𝑇_𝜙∈𝑆_𝑝, 1≤𝑝≤∞ if and only if 𝜙(𝑧)∈𝐿^𝑝(𝔻,𝑑𝜆). In this case, ‖𝑇_𝜙‖²_𝑝≤‖𝑇_𝜙1‖²_𝑝+‖𝑇_𝜙2‖²_𝑝 if 2≤𝑝<∞ and ‖𝑇_𝜙‖²_𝑝≥‖𝑇_𝜙1‖²_𝑝+‖𝑇_𝜙2‖²_𝑝 if 1≤𝑝≤2.

Proof. Suppose 2≤𝑝≤∞ and 𝑇_𝜙∈𝑆_𝑝. Notice that 𝑇_𝜙1=(𝑇_𝜙+𝑇^∗_𝜙)/2 and 𝑇_𝜙2=(𝑇_𝜙−𝑇^∗_𝜙)/2𝑖 and since 𝑆_𝑝 is a Banach space and is closed under adjoints, hence 𝑇_𝜙1 and 𝑇_𝜙2 belong to 𝑆_𝑝. From [1], it follows that 𝜙₁,𝜙₂∈𝐿^𝑝(𝔻,𝑑𝜆). Hence 𝜙=𝜙₁+𝑖𝜙₂∈𝐿^𝑝(𝔻,𝑑𝜆). Now suppose 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 2≤𝑝≤∞. This implies 𝜙₁,𝜙₂∈𝐿^𝑝(𝔻,𝑑𝜆). From [1], it follows that 𝑇_𝜙1,𝑇_𝜙2∈𝑆_𝑝. Hence 𝑇_𝜙∈𝑆_𝑝 as 𝑆_𝑝 is a vector space and ‖𝑇_𝜙‖²_𝑝≤‖𝑇_𝜙1‖²_𝑝+‖𝑇_𝜙2‖²_𝑝 (see [6]). Now let 1≤𝑝<2 and assume 𝑇_𝜙∈𝑆_𝑝. This implies 𝑇_𝜙1,𝑇_𝜙2∈𝑆_𝑝 and therefore by [1], 𝜙₁,𝜙₂∈𝐿^𝑝(𝔻,𝑑𝜆). Hence 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) and by [6], ‖𝑇_𝜙‖²_𝑝≥‖𝑇_𝜙1‖²_𝑝+‖𝑇_𝜙2‖²_𝑝. Now suppose 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 1≤𝑝<2. Then 𝜙₁,𝜙₂∈𝐿^𝑝(𝔻,𝑑𝜆) and hence 𝑇_𝜙1,𝑇_𝜙2∈𝑆_𝑝. Since 𝑆_𝑝 is a vector space, 𝑇_𝜙∈𝑆_𝑝.

Example 2.2. Let 𝜙(𝑧)=(1−|𝑧|)^𝑟, 𝑟>0. Then 𝑇_𝜙∈𝑆₂ only if 𝑟>1/2, and 𝑇_𝜙∈𝑆₁ only if 𝑟>1. This can be seen as follows. The matrix of 𝑇_𝜙 with respect to the standard orthonormal basis {𝑒_𝑛(𝑧)}_𝑛≥0={√𝑛+1𝑧^𝑛}_𝑛≥0 of 𝐿²_𝑎(𝔻) is diagonal, 𝑡_𝑘𝑘=⟨𝑇_𝜙𝑒_𝑘,𝑒_𝑘⟩=(2𝑘+2)!/((𝑟+1)(𝑟+2)⋯(𝑟+2𝑘+2)) and ∑^∞_𝑘=0|𝑡_𝑘𝑘|²=∞ for 𝑟≤1/2. Similarly ∑^∞_𝑘=0𝑡_𝑘𝑘<∞ if 𝑟>1. If 𝑟≤1, ∑^∞_𝑘=0𝑡_𝑘𝑘=∞.

Proposition 2.3. Let 𝜙∈𝐿^∞(𝔻) and suppose that 𝜙 is not the zero function. If 𝑇_𝜙 is compact then Range𝑇_𝜙 is not closed.

Proof. Since 𝑇_𝜙 is compact, hence Range𝑇_𝜙 contains no closed infinite-dimensional subspace of 𝐿²_𝑎(𝔻). If now Range𝑇_𝜙 is closed then Range𝑇_𝜙 is finite dimensional. That is, 𝑇_𝜙 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 that 𝐴 is a positive operator on a Hilbert space 𝐻 and 𝑥 is unit vector in 𝐻, then (i) ⟨𝐴^𝑝𝑥,𝑥⟩≥⟨𝐴𝑥,𝑥⟩^𝑝 for all 𝑝≥1; (ii) ⟨𝐴^𝑝𝑥,𝑥⟩≤⟨𝐴𝑥,𝑥⟩^𝑝 for all 0<𝑝≤1. For proof see [1].

Suppose that 𝜑∶𝔻→𝔻 is analytic. Define the composition operator 𝐶_𝜑 from 𝐿²_𝑎(𝔻) into itself as 𝐶_𝜑𝑓=𝑓∘𝜑. It is shown in [1] that 𝐶_𝜑 is a bounded linear operator on 𝐿²_𝑎(𝔻) and ‖𝐶_𝜑‖≤(1+|𝜑(0)|)/(1−|𝜑(0)|). Given 𝑎∈𝔻 and 𝑓 any measurable function on 𝔻, we define the function 𝐶_𝑎𝑓 by 𝐶_𝑎𝑓(𝑧)=𝑓(𝜙_𝑎(𝑧)), where 𝜙_𝑎∈Aut(𝔻). The map 𝐶_𝑎 is a composition operator on 𝐿²_𝑎(𝔻).

Proposition 2.4. If 𝜙∈𝐿^∞(𝔻) then 𝑇_𝜙 is compact if and only if 𝑇_{𝜙∘𝜙𝑧} is compact.

Proof. This follows from the fact that 𝑇_{𝜙∘𝜙𝑧}=𝑈_𝑧𝑇_𝜙𝑈_𝑧 and as 𝑈²_𝑧=𝐼.

Proposition 2.5. For 𝜙∈𝐿^∞(𝔻), MO(𝜙)²(𝑧)=|𝜙|²(𝑧)−|𝜙(𝑧)|²≤‖𝐻_𝜙𝑘_𝑧‖²+‖𝐻_𝜙𝑘_𝑧‖².

Proof. Observe that ‖𝐻_𝜙𝑘_𝑧‖=‖(𝐼−𝑃)(𝜙𝑘_𝑧)‖=‖(𝐼−𝑃)𝑈_𝑧(𝜙∘𝜙_𝑧)‖=‖𝑈_𝑧(𝐼−𝑃)(𝜙∘𝜙_𝑧)‖=‖(𝐼−𝑃)(𝜙∘𝜙_𝑧)‖=‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖. (2.1) Similarly, we have ‖𝐻_𝜙𝑘_𝑧‖=‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖=‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖. Since 𝜙(𝑧)=𝑃(𝜙∘𝜙_𝑧)(0) and 𝑃𝑔(𝑧)=𝑔(0) for any 𝑔∈𝐿²_𝑎 and all 𝑧∈𝔻, we have MO(𝜙)²(𝑧)=|𝜙|²(𝑧)−|𝜙(𝑧)|²=‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)(0)‖²=‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖²+‖𝑃(𝜙∘𝜙_𝑧)−𝑃(𝜙∘𝜙_𝑧)(0)‖²=‖𝐻_𝜙𝑘_𝑧‖²+‖𝑃(𝜙∘𝜙_𝑧)−𝑃(𝜙∘𝜙_𝑧)(0)‖²=‖𝐻_𝜙𝑘_𝑧‖²+‖𝑃(𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧))‖²≤‖𝐻_𝜙𝑘_𝑧‖²+‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖²=‖𝐻_𝜙𝑘_𝑧‖²+‖𝐻_𝜙𝑘_𝑧‖². (2.2)

Let ℎ>1. The generalized Kantorvich constant 𝐾(𝑝) is defined by ℎ𝐾(𝑝)=^𝑝−ℎ((𝑝−1)(ℎ−1)𝑝−1𝑝ℎ^𝑝−1ℎ^𝑝)−ℎ^𝑝 (2.3) 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: 𝐾(𝑝)⟨𝐴𝑥,𝑥⟩^𝑝≥⟨𝐴^𝑝𝑥,𝑥⟩≥⟨𝐴𝑥,𝑥⟩^𝑝forany𝑝>1orany𝑝<0;⟨𝐴𝑥,𝑥⟩^𝑝≥⟨𝐴^𝑝𝑥,𝑥⟩≥𝐾(𝑝)⟨𝐴𝑥,𝑥⟩^𝑝forany𝑝∈(0,1]. (2.4)(2.5)

The Kantorvich constant 𝐾(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)≥2ℎ^1/4/(ℎ^1/2+1) for 𝑝∈[0,1].

Proposition 2.6. Let 𝑇_𝜙 be strictly positive satisfying 𝑀𝐼≥𝑇_𝜙≥𝑚𝐼>0, where 𝑀>𝑚>0. The following hold.(i)If 0<𝑝<∞ and 𝑇_𝜙∈𝑆_𝑝 then 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆).(ii)If 0<𝑝≤1, 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then 𝑇_𝜙∈𝑆_𝑝.(iii)Let 𝑝∈[1,∞) be such that 𝐾(𝑝)<∞. If 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then 𝑇_𝜙∈𝑆_𝑝.

Proof. Suppose 𝑝>1 and 𝑇_𝜙∈𝑆_𝑝. Then _𝔻⟨𝑇^𝑝_𝜙𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)=_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞. (2.6) Hence by (2.4), ∫_𝔻⟨𝑇_𝜙𝑘_𝑧,𝑘_𝑧⟩^𝑝𝑑𝜆(𝑧)<∞. That is, 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆). Suppose 0<𝑝≤1 and 𝑇_𝜙∈𝑆_𝑝. Then ∫_𝔻⟨𝑇^𝑝_𝜙𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)=∫_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞. Hence from (2.5), it follows that 𝐾(𝑝)∫_𝔻⟨𝑇_𝜙𝑘_𝑧,𝑘_𝑧⟩^𝑝𝑑𝜆(𝑧)<∞. Since 𝐾(𝑝)∈(0,1] for 𝑝∈[0,1], hence 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆).
Now assume 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆). Then if 0<𝑝≤1 then by (2.5), we have ∫_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞ and hence 𝑇_𝜙∈𝑆_𝑝. If 1≤𝑝<∞, then by (2.4) and (2.5), if 𝐾(𝑝)<∞ and 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then ∫_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞ and 𝑇_𝜙∈𝑆_𝑝.

3. Schatten Class Operators

In this section, we will obtain conditions to describe Schatten class Toeplitz operators on Bergman space 𝐿²_𝑎(𝔻). The results of this paper hold for Bergman spaces 𝐿²_𝑎(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.

Let 𝐵𝑇={𝑓∈𝐿¹∶‖𝑓‖_𝐵𝑇=sup_𝑧∈𝔻|𝑓|(𝑧)<∞}. The space 𝐿^∞ is properly contained in 𝐵𝑇 (see [9]) and if 𝜙∈𝐵𝑇 then 𝑇_𝜙 is bounded on 𝐿²_𝑎 and there is a constant 𝐶 such that ‖𝑇_𝜙‖≤𝐶‖𝜙‖_𝐵𝑇.

Theorem 3.1. Suppose 1≤𝑝<∞ and 𝑑𝜆(𝑧)=𝑑𝐴(𝑧)/(1−|𝑧|²)². Then the following hold. (1) If 𝑇_𝜙∈𝑆_𝑝, then 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆). (2) If 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) and 𝑇_𝜙∈𝑆_𝑝.

Proof. Suppose 𝑇_𝜙∈𝑆_𝑝. Then _𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑤,𝑘_𝑤⟩𝑑𝜆(𝑤)<∞. (3.1) That is, ∫_𝔻⟨(𝑇^∗_𝜙𝑇_𝜙)^𝑝/2𝑘_𝑤,𝑘_𝑤⟩𝑑𝜆(𝑤)<∞. If 2≤𝑝<∞, then _𝔻⟨𝑇^∗_𝜙𝑇_𝜙𝑘_𝑤,𝑘_𝑤⟩^𝑝/2𝑑𝜆(𝑤)≤_𝔻⟨(𝑇^∗_𝜙𝑇_𝜙)^𝑝/2𝑘_𝑤,𝑘_𝑤⟩𝑑𝜆(𝑤)<∞. (3.2) This implies _𝔻‖𝑃(𝜙∘𝜙_𝑤)‖^𝑝𝑑𝜆(𝑤)=_𝔻‖𝑃(𝑈_𝑤(𝜙𝑘_𝑤))‖^𝑝=𝑑𝜆(𝑤)_𝔻‖𝑈_𝑤𝑇_𝜙𝑘_𝑤‖^𝑝𝑑𝜆(𝑤)=_𝔻‖𝑇_𝜙𝑘_𝑤‖^𝑝=𝑑𝜆(𝑤)_𝔻⟨𝑇^∗_𝜙𝑇_𝜙𝑘_𝑤,𝑘_𝑤⟩^𝑝/2𝑑𝜆(𝑤)<∞. (3.3) Now |𝑃(𝜙∘𝜙_𝑤)(0)|=|⟨𝑃(𝜙∘𝜙_𝑤),1⟩|=|⟨𝑈_𝑤(𝑇_𝜙𝑘_𝑤),1⟩|=|⟨𝑇_𝜙𝑘_𝑤,𝑈_𝑤1⟩|=|⟨𝑇_𝜙𝑘_𝑤,𝑘_𝑤⟩|≤‖𝑇_𝜙𝑘_𝑤‖=‖𝑃(𝜙∘𝜙_𝑤)‖. Thus ∫_𝔻|𝑃(𝜙∘𝜙_𝑤)(0)|^𝑝𝑑𝜆(𝑤)<∞. That is, ∫_𝔻|𝜙(𝑤)|^𝑝𝑑𝜆(𝑤)<∞ and 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆). Suppose 1≤𝑝<2. Then by Heinz inequality [10], it follows that ∞>_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑤,𝑘_𝑤=⟩𝑑𝜆(𝑤)_𝔻⟨|𝑇_𝜙|^2⋅𝑝/2𝑘_𝑤,𝑘_𝑤⟩𝑑𝜆(𝑤)≥_𝔻|⟨𝑇_𝜙𝑘_𝑤,𝑘_𝑤⟩|²⟨|𝑇^∗_𝜙|^{2(1−𝑝/2)}𝑘_𝑤,𝑘_𝑤⟩=𝑑𝜆(𝑤)_𝔻|𝜙(𝑤)|²‖𝑃(𝜙∘𝜙_𝑤)‖^2−𝑝=𝑑𝜆(𝑤)_𝔻|𝜙(𝑤)|²‖𝑃(𝜙∘𝜙_𝑤)‖^𝑝−2≥𝑑𝜆(𝑤)_𝔻|𝜙(𝑤)|²‖𝑃(𝜙∘𝜙_𝑤)‖²‖𝑃(𝜙∘𝜙_𝑤)‖^𝑝≥𝑑𝜆(𝑤)_𝔻|𝜙(𝑤)|²𝐶²‖𝜙‖²_𝐵𝑇|𝑃(𝜙∘𝜙_𝑤)(0)|^𝑝=𝑑𝜆(𝑤)_𝔻|𝜙(𝑤)|²𝐶²‖𝜙‖²_𝐵𝑇|𝜙(𝑤)|^𝑝𝑑𝜆(𝑤), (3.4) since ⟨|𝑇^∗_𝜙|^2−𝑝𝑘_𝑤,𝑘_𝑤⟩=⟨|𝑇^∗_𝜙|^{2⋅(2−𝑝)/2}𝑘_𝑤,𝑘_𝑤⟩≤⟨|𝑇^∗_𝜙|²𝑘_𝑤,𝑘_𝑤⟩^(2−𝑝)/2=⟨𝑇_𝜙𝑇^∗_𝜙𝑘_𝑤,𝑘_𝑤⟩^(2−𝑝)/2=‖𝑇^∗_𝜙𝑘_𝑤‖^2−𝑝=‖𝑃(𝜙∘𝜙_𝑤)‖^2−𝑝. (3.5)
Hence _𝔻|𝜙(𝑤)|^𝑝+2𝑑𝜆(𝑤)<∞, (3.6) and therefore ∫_𝔻|𝜙(𝑤)|^𝑝𝑑𝜆(𝑤)<∞. Thus 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆).
Now suppose 𝜙∈𝐿¹(𝔻,𝑑𝜆). Then the change of the order of integration _𝔻|𝜙(𝑤)|𝑑𝜆(𝑤)=_𝔻|𝜙(𝑤)|𝑑𝐴(𝑤)(1−|𝑤|²)²≤_𝔻(_𝔻|𝜙(𝑧)|(1−|𝑤|²)²|1−𝐰𝑧|⁴𝑑𝐴(𝑧))𝑑𝐴(𝑤)(1−|𝑤|²)²=_𝔻|𝜙(𝑧)|_𝔻𝑑𝐴(𝑤)|1−𝐰𝑧|⁴𝑑𝐴(𝑧)=_𝔻|𝜙(𝑧)|⟨𝑘_𝑧,𝑘_𝑧=⟩𝑑𝐴(𝑧)_𝔻|𝜙(𝑧)|𝑑𝐴(𝑧)(1−|𝑧|²)² (3.7) is justified by the positivity of the integrand. Hence 𝜙∈𝐿¹(𝔻,𝑑𝜆). Similarly if 𝜙∈𝐿^∞(𝔻) then 𝜙∈𝐿^∞(𝔻) as |𝜙(𝑤)|=|⟨𝜙𝑘_𝑤,𝑘_𝑤⟩|≤‖𝜙𝑘_𝑤‖₂‖𝑘_𝑤‖₂≤‖𝜙‖_∞‖𝑘_𝑤‖²₂=‖𝜙‖_∞. By Marcinkiewicz interpolation theorem it follows that if 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) for 1≤𝑝≤∞. Now suppose 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 1≤𝑝≤∞. We will prove 𝑇_𝜙∈𝑆_𝑝. The case 𝑝=+∞ is trivial. By interpolation we need only to prove the result for 𝑝=1. Suppose 𝜙∈𝐿¹(𝔻,𝑑𝜆) and {𝑒_𝑛}={√𝑛+1𝑧^𝑛}^∞_𝑛=0 is the standard orthonormal basis for 𝐿²_𝑎(𝔻). Now ⟨𝑇_𝜙𝑒_𝑛,𝑒_𝑛⟩=∫_𝔻|𝑒_𝑛(𝑧)|²𝜙(𝑧)𝑑𝐴(𝑧) and ^∞_𝑛=0|⟨𝑇_𝜙𝑒_𝑛,𝑒_𝑛⟩|≤_𝔻^∞_𝑛=0|𝑒_𝑛(𝑧)|²≤|𝜙(𝑧)|𝑑𝐴(𝑧)_𝔻=𝐾(𝑧,𝑧)|𝜙(𝑧)|𝑑𝐴(𝑧)_𝔻|𝜙(𝑧)|𝑑𝜆(𝑧). (3.8) Thus 𝑇_𝜙∈𝑆₁ and ‖𝑇_𝜙‖_𝑆1≤∫_𝔻|𝜙(𝑧)|𝑑𝜆(𝑧).

It is not so difficult to verify the conditions in Theorem 3.1.

Example 3.2. Let Φ(𝑧)=(1−|𝑧|²)log(1−|𝑧|²). Then Φ∈𝐿²(𝔻,𝑑𝜆). This can be verified as follows: _𝔻(1−|𝑧|²)²|log(1−|𝑧|²)|²=𝑑𝜆(𝑧)_𝔻(1−|𝑧|²)²|log(1−|𝑧|²)|²𝑑𝐴(𝑧)|(1−|𝑧|²)|²=_𝔻|log(1−|𝑧|²)|²=𝑑𝐴(𝑧)₁₀|log(1−𝑟²)|²=2𝑟𝑑𝑟₁₀|log(1−𝑡)|²=𝑑𝑡₁₀|log𝑡|²𝑑𝑡, (3.9) and changing the variable to 𝑦=−log𝑡, this reduces to ∫^∞₀𝑦²𝑒^−𝑦𝑑𝑦=Γ(3)=2<∞. Thus 𝑇_Φ∈𝑆₂.

Example 3.3. Let 𝑔(𝑧)=ln|𝑧|². Then _𝔻|𝑔(𝑧)|²𝑑𝜆(𝑧)=¹₀(ln𝑡)1−𝑡²=𝑑𝑡_{10∞∞𝑛=0𝑚=0}𝑡^𝑚+𝑛ln²=𝑡𝑑𝑡_{∞∞𝑛=0𝑚=0}2(𝑚+𝑛+1)³=^∞_𝑘=02(𝑘+1)²=𝜋²3<∞. (3.10) Thus 𝑔∈𝐿²(𝔻,𝑑𝜆). Direct computation reveals that 𝑔(𝑥)=|𝑥|²−1 and 𝑔∈𝐿²(𝔻,𝑑𝜆).

Example 3.4. Let 𝜙(𝑧)=1−|𝑧|. Then by Example 2.2, 𝑇_𝜙∈𝑆₂ and _𝔻|𝜙(𝑧)|²𝑑𝜆(𝑧)=_𝔻(1−|𝑧|)²𝑑𝐴(𝑧)(1−|𝑧|²)²=_𝔻𝑑𝐴(𝑧)(1+|𝑧|)²≤_𝔻𝑑𝐴(𝑧)=1. (3.11) Thus 𝜙∈𝐿²(𝔻,𝑑𝜆) and hence 𝜙∈𝐿²(𝔻,𝑑𝜆).
It may be noted that the space 𝐿^𝑝(𝔻,𝑑𝜆), 1≤𝑝<∞, contains no nonzero harmonic functions and even no nonzero constants. To see this, for example, for 𝐿²(𝔻,𝑑𝜆), let 1𝑀(𝑟)=2𝜋₀^2𝜋|𝑓(𝑟𝑒^𝑖𝑡)|²𝑑𝑡. (3.12) This is a nonnegative and nondecreasing function of 𝑟 and ‖𝑓‖²_{𝐿2(𝔻,𝑑𝜆)}=∫¹₀𝑀(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 in 𝐿²(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 𝑓(𝑤)=∫_𝔻𝑓(𝜙_𝑤(𝑧))𝑑𝐴(𝑤)≥𝑓(𝜙_𝑤(0))=𝑓(𝑤). Hence 𝑓≥𝑓. Let Δ_ℎ∶=(1−|𝑧|²)²(𝜕²/𝜕𝑧𝜕𝐳). It can be verified that Δ_ℎ(𝑓∘𝜙_𝑎)=(Δ_ℎ𝑓)∘𝜙_𝑎 and note that Δ_ℎ𝑓=Δ_ℎ𝑓≥0 since Δ_ℎ𝑓≥0. In other words, 𝑓 is also subharmonic. Proceeding by induction, if we define 𝐵𝑔=𝑔 on 𝐿²(𝔻,𝑑𝜆) then we obtain 𝐵^𝑛𝑓 is subharmonic for all 𝑛∈ℕ and {𝐵^𝑛𝑓}_𝑛∈ℕ is a nondecreasing sequence of functions.

Corollary 3.5. If 𝜙∈ℎ^∞(𝔻), the space of bounded harmonic functions on 𝔻 and 1≤𝑝<∞, then 𝑇_𝜙∈𝑆_𝑝 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 𝜙=𝜙.

Corollary 3.6. If 𝜙 is a real-valued bounded subharmonic function on 𝔻, 1≤𝑝<∞, then 𝑇_𝜙∈𝑆_𝑝 if and only if 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆).

Proof. By Theorem 3.1, if 𝑇_𝜙∈𝑆_𝑝 then 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆). Conversely if 𝜙 is real-valued, subharmonic, bounded on 𝔻 and 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then 𝜙 is also subharmonic and the subharmonicity of 𝜙 implies that 𝜙(𝑤)=∫_𝔻𝜙(𝜙_𝑤(𝑧))𝑑𝐴(𝑧)≥𝜙(𝜙_𝑤(0))=𝜙(𝑤). Hence ∫_𝔻|𝜙(𝑤)|^𝑝𝑑𝜆(𝑤)=∫_𝔻|𝜙(𝑤)|^𝑝𝑑𝜆(𝑤) as 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) implies 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) and the result follows from Theorem 3.1.

Corollary 3.7 follows immediately from Corollary 3.5. We present a proof of Corollary 3.7 to show a different method of approach.

Corollary 3.7. If 𝜙∈𝐻^∞(𝔻)∪𝐻^∞(𝔻) then the Toeplitz operator 𝑇_𝜙∈𝑆_𝑝,1≤𝑝<∞ if and only if 𝜙≡0.

Proof. Notice that if 𝜙∈𝐻^∞(𝔻)∪𝐻^∞(𝔻) then the following two possibilities hold.(∗) For all 𝑧∈𝔻, either 𝑃(𝜙∘𝜙_𝑧)≠𝑃(𝜙∘𝜙_𝑧)(0) or 𝑃(𝜙∘𝜙_𝐳)≠𝑃(𝜙∘𝜙_𝐳)(0).(∗∗) There exists 𝑧∈𝐷 such that 𝑃(𝜙∘𝜙_𝑧)=𝑃(𝜙∘𝜙_𝑧)(0) and 𝑃(𝜙∘𝜙_𝐳)=𝑃(𝜙∘𝜙_𝐳)(0).
Suppose that (∗∗) holds and let 𝑧∈𝔻 be such that 𝑃(𝜙∘𝜙_𝑧)=𝑃(𝜙∘𝜙_𝑧)(0) and 𝑃(𝜙∘𝜙_𝐳)=𝑃(𝜙∘𝜙_𝐳)(0). Then this implies 𝑇_𝜙𝑘_𝑧=𝑐_𝑧𝑘_𝑧 and 𝑇_𝜙𝑘_𝑧=𝐜_𝐳𝑘_𝑧. Hence 𝑇_𝜙 has an eigenvalue. From [1], it follows that 𝜙 is a constant and 𝑇_𝜙 is not compact unless 𝜙≡0.
Now suppose that (∗) holds, 1≤𝑝<∞. Then for all 𝑧∈𝔻, ‖𝐻_𝜙𝑘_𝑧‖²+‖𝐻_𝜙𝑘_𝑧‖²=‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖²+‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)‖²<‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)(0)‖²+‖𝜙∘𝜙_𝑧−𝑃(𝜙∘𝜙_𝑧)(0)‖²=2(|𝜙|²(𝑧)−|𝜙(𝑧)|²). Let 𝑐_𝑧=(‖𝐻_𝜙𝑘_𝑧‖²+‖𝐻_𝜙𝑘_𝑧‖²)/(|𝜙|²(𝑧)−|𝜙(𝑧)|²). Note that 1≤𝑐_𝑧<2 for all 𝑧∈𝔻. It follows from Proposition 2.5 and from the previous discussion.
Notice 𝑇_𝜙 compact implies that 𝑇^∗_𝜙𝑇_𝜙=𝑇_|𝜙|²−𝐻^∗_𝜙𝐻_𝜙 is compact. Thus 0≤⟨𝑇^∗_𝜙𝑇_𝜙𝑘_𝑧,𝑘_𝑧⟩=|𝜙|²(𝑧)−‖𝐻_𝜙𝑘_𝑧‖²→0 as |𝑧|→1⁻. Similarly since 𝑇_𝜙𝑇^∗_𝜙 is compact, 0≤|𝜙|²(𝑧)−‖𝐻_𝜙𝑘_𝑧‖²→0 as |𝑧|→1⁻.
Thus 0≤2|𝜙|²(𝑧)−(‖𝐻_𝜙𝑘_𝑧‖²+‖𝐻_𝜙𝑘_𝑧‖²)→0 as |𝑧|→1⁻. Hence it follows that 0≤2|𝜙|²(𝑧)−𝑐_𝑧(|𝜙|²(𝑧)−|𝜙(𝑧)|²)=2|𝜙|²(𝑧)−(‖𝐻_𝜙𝑘_𝑧‖²+‖𝐻_𝜙𝑘_𝑧‖²)→0 as |𝑧|→1⁻. So 0≤(2−𝑐_𝑧)|𝜙|²(𝑧)+𝑐_𝑧|𝜙(𝑧)|²→0 as |𝑧|→1⁻, where 1≤𝑐_𝑧<2. Since 𝑇_𝜙 is compact, |𝜙(𝑧)|→0 as |𝑧|→1⁻. Thus |𝜙|²(𝑧)→0 as |𝑧|→1⁻. Similarly since 𝑇_{𝜙∘𝜙𝑧} is compact we can show that |𝜙∘𝜙_𝑧|²(𝑧)→0 as |𝑧|→1⁻. Thus _𝔻|(𝜙∘𝜙_𝑧)(𝑤)|²|𝑘_𝑧(𝑤)|²𝑑𝐴(𝑤)→0 (3.13) as |𝑧|→1⁻. Hence ∫_𝔻|𝜙(𝑤)|²𝑑𝐴(𝑤)=0 as 𝑈_𝑧𝑘_𝑧=𝑘_𝑧(𝜙_𝑧(𝑤))𝑘_𝑧(𝑤)=1. It follows therefore that 𝜙(𝑤)=0 almost everywhere and hence 𝜙≡0. Therefore, 𝑇_𝜙≡0.

Let 𝑇(𝑧)=⟨𝑇𝑘_𝑧,𝑘_𝑧⟩ for 𝑇∈ℒ(𝐿²_𝑎(𝔻)). It follows from Heinz inequality [10] that |⟨𝑇_𝜙𝑘_𝑧,𝑘_𝑧⟩|²≤⟨|𝑇_𝜙|^2𝛼𝑘_𝑧,𝑘_𝑧⟩⟨|𝑇^∗_𝜙|^2(1−𝛼)𝑘_𝑧,𝑘_𝑧⟩ (3.14) for all 𝑧∈𝔻 and 0≤𝛼≤1. That is, |𝜙(𝑧)|²≤|𝑇_𝜙|^2𝛼(𝑧)|𝑇^∗_𝜙|^2(1−𝛼)(𝑧) for all 𝑧∈𝔻. Hence if |𝑇_𝜙|^2𝛼∈𝐿^𝑝(𝔻,𝑑𝜆) and |𝑇^∗_𝜙|^2(1−𝛼)∈𝐿^𝑞(𝔻,𝑑𝜆), 1/𝑝+1/𝑞=1, it follows from Holders inequality that 𝜙∈𝐿²(𝔻,𝑑𝜆). From Theorem 3.1, it follows that 𝑇_𝜙∈𝑆₂.

We say 𝑇 majorizes 𝑆∈ℒ(𝐿²_𝑎(𝔻)) if ‖𝑆𝑓‖≤𝑀‖𝑇𝑓‖ for all 𝑓∈𝐿²_𝑎(𝔻).

Corollary 3.8. If 𝜙≥0, 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 1≤𝑝≤2, and 𝑇_𝜙 majorizes 𝑇_𝜓 then 𝑇_𝜓∈𝑆_𝑝.

Proof. Since 𝑇_𝜙 majorizes 𝑇_𝜓, it follows that ‖𝑇_𝜓𝑓‖≤𝑀‖𝑇_𝜙𝑓‖ for some 𝑀>0 and for all 𝑓∈𝐿²_𝑎. Hence ⟨𝑇^∗_𝜓𝑇_𝜓𝑓,𝑓⟩≤𝑀²⟨𝑇^∗_𝜙𝑇_𝜙𝑓,𝑓⟩ for all 𝑓∈𝐿²_𝑎. That is, 𝑇^∗_𝜓𝑇_𝜓≤𝑀²𝑇^∗_𝜙𝑇_𝜙. Since 1≤𝑝≤2, we obtain from [10, 11] that (𝑇^∗_𝜓𝑇_𝜓)^𝑝/2≤𝑀^𝑝(𝑇^∗_𝜙𝑇_𝜙)^𝑝/2. That is, |𝑇_𝜓|^𝑝≤𝑀^𝑝|𝑇_𝜙|^𝑝 for 1≤𝑝≤2. Now if 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 𝜙≥0, then by [1], 𝑇_𝜙∈𝑆_𝑝 and ∫_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞. Hence ∫_𝔻⟨|𝑇_𝜓|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞ and 𝑇_𝜓∈𝑆_𝑝.

Corollary 3.9. Let 𝜙,𝜓∈𝐿^∞(𝔻). If Range𝑇_𝝍⊆Range𝑇_𝜙 and 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 1≤𝑝≤2 then 𝑇_𝜓∈𝑆_𝑝 and 𝜓∈𝐿^𝑝(𝔻,𝑑𝜆).

Proof. Range𝑇_𝝍⊆Range𝑇_𝜙 implies that 𝑇_𝜙 majorizes 𝑇_𝜓. Hence ‖𝑇_𝜓𝑓‖≤𝑀‖𝑇_𝜙𝑓‖ for some 𝑀>0 and for all 𝑓∈𝐿²_𝑎. Hence ⟨𝑇^∗_𝜓𝑇_𝜓𝑓,𝑓⟩≤𝑀²⟨𝑇^∗_𝜙𝑇_𝜙𝑓,𝑓⟩ for all 𝑓∈𝐿²_𝑎. That is, 𝑇^∗_𝜓𝑇_𝜓≤𝑀²𝑇^∗_𝜙𝑇_𝜙. Since 1≤𝑝≤2, we obtain from [10, 11] that (𝑇^∗_𝜓𝑇_𝜓)^𝑝/2≤𝑀^𝑝(𝑇^∗_𝜙𝑇_𝜙)^𝑝/2. That is, |𝑇_𝜓|^𝑝≤𝑀^𝑝|𝑇_𝜙|^𝑝 for 1≤𝑝≤2. Now if 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) then 𝑇_𝜙∈𝑆_𝑝 and ∫_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞. Hence ∫_𝔻⟨|𝑇_𝜓|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞. Thus 𝑇_𝜓∈𝑆_𝑝 and 𝜓∈𝐿^𝑝(𝔻,𝑑𝜆).

Definition 3.10. A function 𝐺∈𝐿²_𝑎(𝐷)(𝐺∈𝐻²) is called an inner function [12] in 𝐿²_𝑎(𝐷) (resp., 𝐻²) if |𝐺|²−1 is orthogonal to 𝐻^∞. If for some inner function 𝐺∈𝐿²_𝑎(𝐷) the following conditions hold then we call it a finite zero divisor in 𝐿²_𝑎(𝐷). (i) 𝐺 vanishes on 𝐚={𝑎_𝑗}^𝑁_𝑗=1, 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,𝑎_𝑛) and certain of their derivatives. (iv) |𝐺|²−1 is orthogonal to 𝐿¹_ℎ, the class of harmonic functions in 𝐿¹ of the disc.

Corollary 3.11. Suppose 𝜙∈𝐿^∞(𝔻) and ker𝑇_𝜙⊆𝐺𝐿²_𝑎, where 𝐺 is a finite inner divisor in 𝐿²_𝑎(𝔻). If 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆),1≤𝑝≤2, then 𝑆_𝜓∈𝑆_𝑝, where 𝑆_𝜓 is the little Hankel operator on 𝐿²_𝑎(𝔻) such that ker𝑆_𝜓=𝐺𝐿²_𝑎.

Proof. Let 𝐛=(𝑏_𝑗)^𝑁_𝑗=1 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∈𝐿²_𝑎(𝐷)∶𝑓=0on𝐛}. Then 𝐺 is the solution of the extremal problem sup{Re𝑓^(𝑛)(0)∶𝑓∈ℐ,‖𝑓‖_𝐿²≤1}, (3.15) 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 𝜓=^𝑁𝑚_𝑗=1^𝑗−1_𝜈=0𝑐_𝑗𝜈𝜕^𝜈𝜕𝑏_𝑗^𝜈𝐾_𝑏𝑗(𝑧), (3.16) where 𝑐_𝑗𝜈≠0 for all 𝑗,𝜈 and 𝑚_𝑗 is the number of times 𝑏_𝑗 appear in b, ker𝑆_𝜓=𝐺𝐿²_𝑎(𝐷). Now ker𝑇_𝜙⊆𝐺𝐿²_𝑎=ker𝑆_𝜓 implies [13] the operator 𝑇_𝜙 that majorizes 𝑆_𝜓. Since 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆), 1≤𝑝≤2, 𝑇_𝜙∈𝑆_𝑝. By similar arguments as in Corollary 3.9 one can show that 𝑆_𝜓∈𝑆_𝑝.

For any 𝑎∈𝔻, let 𝛾_𝑎 be the unique geodesic such that 𝛾_𝑎(0)=0, 𝛾_𝑎(1)=𝑎. Then there exists a unique 𝜙_𝑎∈Aut(𝔻) such that 𝜙_𝑎∘𝜙_𝑎(𝑧)≡𝑧 and 𝛾_𝑎(1/2) is an isolated fixed point of 𝜙_𝑎. Further 𝜙_𝑎 is the geodesic symmetry at 𝛾_𝑎(1/2). We denote by 𝑚_𝑎 the geodesic midpoint 𝛾_𝑎(1/2) of 0 and 𝑎. It can easily be verified that each 𝜙_𝑎 has 𝑚_𝑎 as a unique fixed point and for 𝑎∈𝔻, 𝑚_𝑎=((1−1−|𝑎|²)/|𝑎|²)𝑎.

Given 𝜆∈𝔻 and a measurable function 𝑓 on 𝔻, we have 𝑓∘𝜙_𝜆=𝑓 if and only if there exists an even function 𝑔 on 𝔻 such that 𝑓=𝑔∘𝜙_𝑚𝜆; 𝑓∘𝜙_𝜆=−𝑓 if and only if there exists an odd function 𝑔 on 𝔻 such that 𝑓=𝑔∘𝜙_𝑚𝜆. For proof of this fact see [14].

Corollary 3.12. If 𝜙=𝜃∘𝜙_𝑚𝑧 for some 𝑧∈𝔻, 𝜃∈𝐿^∞(𝔻) and where 𝜃 is an even function and 𝑇_𝜙∈𝑆_𝑝, 1≤𝑝<∞, then ‖𝑇_{𝜓∘𝜙𝑧−𝜓+𝜙}‖_𝑆𝑝≥‖𝑇_𝜙‖_𝑆𝑝 (3.17) for all 𝑇_𝜓∈𝑆_𝑝.

Proof. Since 𝜙=𝜃∘𝜙_𝑚𝑧 where 𝜃 is an even function, by [14], we have 𝜙∘𝜙_𝑧=𝜙. Hence 𝑈_𝑧𝑇_𝜙𝑈_𝑧=𝑇_{𝜙∘𝜙𝑧}=𝑇_𝜙. As 𝑈_𝑧 is unitary, from [15], it follows that ‖𝑈_𝑧𝑇_𝜓𝑈_𝑧−𝑇_𝜓+𝑇_𝜙‖_𝑆𝑝≥‖𝑇_𝜙‖_𝑆𝑝 (3.18) for all 𝑇_𝜓∈𝑆_𝑝. Hence ‖𝑇_{𝜓∘𝜙𝑧−𝜓+𝜙}‖_𝑆𝑝≥‖𝑇_𝜙‖_𝑆𝑝 (3.19) for all 𝑇_𝜓∈𝑆_𝑝.

Theorem 3.13. Suppose there exist constants 𝐶,𝑀>0 such that 𝐶>𝑀 and ‖𝑃(𝜙∘𝜙_𝑧)−𝐶‖≤𝑀 for all 𝑧∈𝔻, 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) and 𝑝∈[1,∞) is such that the Kantorvich constant 𝐾(𝑝)∈(0,∞). If 𝑇_𝜙 is bounded below then 𝑇_𝜙∈𝑆_𝑝.

Proof. If there exist 𝐶>𝑀>0 such that for all 𝑧∈𝔻, we have ‖𝑃(𝜙∘𝜙_𝑧)−𝐶‖≤𝑀 then ‖𝑇_𝜙𝑘_𝑧−𝐶𝑘_𝑧‖=‖𝑈_𝑧𝑃(𝜙∘𝜙_𝑧)−𝐶𝑈_𝑧1‖=‖𝑃(𝜙∘𝜙_𝑧)−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, ‖𝑇_𝜙𝑘_𝑧−((Γ+𝛾)/2)𝑘_𝑧‖≤(1/2)|Γ−𝛾|‖𝑘_𝑧‖. By [16], it follows that there exists 𝐿>0 such that |𝜙(𝑧)|=|⟨𝑇_𝜙𝑘_𝑧,𝑘_𝑧⟩|≥𝐿‖𝑇_𝜙𝑘_𝑧‖ for all 𝑧∈𝔻. Notice that the reproducing kernels {𝑘_𝑧∶𝑧∈𝔻} span 𝐿²_𝑎(𝔻) and 𝑇_𝜙 is bounded below. Hence there exist 𝑟>0, 𝑠>0 such that 𝑠≤𝑇^∗_𝜙𝑇_𝜙≤𝑟. Now if 𝜙∈𝐿^𝑝(𝔻,𝑑𝜆) and 𝑝∈[1,∞) is such that the Kantorvich constant 𝐾(𝑝)∈(0,∞) then ∫_𝔻‖𝑇_𝜙𝑘_𝑧‖^𝑝𝑑𝜆(𝑧)<∞. From inequalities (2.4) and (2.5), it follows that ∫_𝔻⟨|𝑇_𝜙|^𝑝𝑘_𝑧,𝑘_𝑧⟩𝑑𝜆(𝑧)<∞. Hence 𝑇_𝜙∈𝑆_𝑝.

Let 𝐿²(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 ℛ_𝑛 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: 𝐻^∞(𝕋)⊂𝐻^∞(𝕋)+ℛ₁⊂𝐻^∞(𝕋)+ℛ₂⊂⋯𝐻^∞(𝕋)+𝐶(𝕋). (3.20) For 𝜙∈𝐿^∞(𝕋), let 𝐵_𝜙∶𝐻²→𝐻² be the Toeplitz operator on 𝐻² with symbol 𝜙, and let Γ_𝜙∶𝐻²→𝐻² be the Hankel operator with symbol 𝜙. Nehari [17], Adamjan et al. [5], and Hartman [18], respectively, proved the following results:

(i)‖Γ_𝜙‖=𝑑(𝜙,𝐻^∞(𝕋));(ii)𝑠_𝑘(Γ_𝜙)=𝑑(𝜙,𝐻^∞(𝕋)+ℛ_𝑛);(iii)‖Γ_𝜙‖_𝑒=𝑑(𝜙,𝐻^∞(𝕋)+𝐶(𝕋)),

where ‖Γ_𝜙‖_𝑒 denotes the essential norm of Γ_𝜙. Feintuch in [19, 20] obtained the following operator theoretic analogues of these distance formulae. Given 𝑇∈ℒ(𝐻²(𝕋)), define a sequence {Γ_𝑛(𝕋)} of operators on 𝐻²(𝕋) by Γ_𝑛(𝕋)=𝐽_𝑛𝑇𝑆^𝑛+1, 𝑛=1,2,…, where for 𝑧∈𝕋, 𝐽_𝑛𝑧^𝑖=𝑧^𝑛−𝑖,if0≤𝑖≤𝑛,0,if𝑖>𝑛, (3.21) and 𝑆 is the unilateral shift on 𝐻²(𝕋). Let 𝒯 denote the family of operators 𝑇 for which {Γ_𝑛(𝕋)} converges strongly and ℒ₀={𝑇∈𝒯∶Γ_𝑛ℒ(𝑇)convergesstronglyto0};_𝑘=ℒ₀+{𝐵_𝜙∶𝜙∈𝐻^∞(𝕋)+ℛ_𝑘ℒ};_∞=ℒ₀+{𝐵_𝜙∶𝜙∈𝐻^∞(𝕋)+𝐶(𝕋)}. (3.22)

Then (i) 𝑑(𝑇,ℒ₀)=‖Γ(𝕋)‖; (ii) 𝑑(𝑇,ℒ_𝑘)=𝑠_𝑘(Γ(𝕋)); (iii) 𝑑(𝑇,ℒ_∞)=‖Γ(𝕋)‖_𝑒. In particular when 𝑇=𝐵_𝜙, 𝜙∈𝐿^∞(𝕋) we have

(i)𝑑(𝐵_𝜙,ℒ₀)=‖Γ_𝜙‖=𝑑(𝜙,𝐻^∞(𝕋));(ii)𝑑(𝐵_𝜙,ℒ_𝑘)=𝑠_𝑘(Γ_𝜙)=𝑑(𝜙,𝐻^∞(𝕋)+ℛ_𝑘);(iii)𝑑(𝐵_𝜙,ℒ_∞)=‖Γ_𝜙‖_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.