Quasinormality and Numerical Ranges of Certain Classes of Dual Toeplitz Operators

Abstract

The study of dual Toeplitz operators was elaborated by Stroethoff and Zheng (2002), where various corresponding algebraic and spectral properties were established. In this paper, we characterize numerical ranges of certain classes of dual Toeplitz operators. Moreover, we introduce the analog of Halmos' fifth classification problem for quasinormal dual Toeplitz operators. In particular, we show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols which are not normal.

1. Introduction

Let 𝔻 be the unit disk of the complex plane ℂ, and let 𝑑𝐴 be the Lebesgue measure on 𝔻. The Lebesgue space of (classes of) square summable complex-valued functions is denoted by ℒ²(𝔻,𝑑𝐴). The Bergman space ℒ²_𝑎 is the Hilbert subspace of ℒ²(𝔻,𝑑𝐴) consisting of analytic functions. The orthogonal complement of ℒ²_𝑎 in ℒ²(𝔻,𝑑𝐴) is denoted by (ℒ²_𝑎)^⟂. The Hilbert space (ℒ²_𝑎)^⟂ is readily seen to be not a reproducing kernel Hilbert space. This is one of the major difficulties that occurs when dealing with this space. A second one is the fact that its elements have no standard common qualities such as analyticity harmonicity, while a lesser difficulty is the complicated form of the corresponding basis.

Despite the difficulties just listed, Stroethoff and Zheng in [1, 2] have adopted new techniques to investigate various properties of a class of operators acting on (ℒ²_𝑎)^⟂, namely, dual Toeplitz operators. A dual Toeplitz operator is defined on (ℒ²_𝑎)^⟂ to be a multiplication (by the symbol) followed by a projection onto (ℒ²_𝑎)^⟂. Although dual Toeplitz operators are different from Toeplitz operators in many respects, they do share some properties with them. But surprisingly, dual Toeplitz operators on (ℒ²_𝑎)^⟂ resemble much more Hardy space Toeplitz operators than Bergman space Toeplitz operators. Lu in [3] and Cheng and Yu in [4] considered dual Toeplitz operators in higher dimensions; while Yu and Wu in [5] considered dual Toeplitz operators in the framework of Dirichlet spaces.

The study of the numerical ranges of Hardy space Toeplitz operators goes back to Brown and Halmos [6]. Subsequent treatment was reconsidered in Halmos' book [7]. Later on, Klein [8] showed that the numerical range depends only on the spectrum of the given Hardy space Toeplitz operator. The Bergman space case was successfully considered only twenty years later by Thukral [9] in case of bounded harmonic symbols. More recently Choe and Lee [10],as well as Gu[11], treat higher-dimensional Bergman space analogs. The case of Bergman space Toeplitz operators with bounded radial symbols has been considered very recently by Wang and Wu [12]. The connection between spectral sets and numerical ranges was considered first by Schreiber [13]. Further investigations had been pursued by Hildebrandt [14] and Clark [15]. The subnormality, and particularly the quasinormality, of Hardy space Toeplitz operators has been discussed chronologically by Itô and Wong [16], Amemiya et al. [17], Abrahamse [18], Cowen and Long [19], Cowen [20, 21], Lee [22], and Yoshino [23]. The case of Bergman space Toeplitz operators has been discussed by Faour in [24] as well as by Jim Gleason in a recent preprint.

Accordingly, in this paper, we mainly investigate qualitative properties of the numerical range of a dual Toeplitz operator. We consider various classes of such operators, such as normal and quasinormal ones. We completely describe the numerical ranges of some of them and establish the main qualitative properties of the numerical ranges of others. We also shed some light on the analog of Halmos' fifth problem on the classification of subnormal Toeplitz operators.

Our paper is organized as follows: in Section 2, we exhibit some preliminary concepts needed in the sequel. Section 3 mainly concerns the description of the numerical ranges of normal dual Toeplitz operators. Section 4 contains the characterization of the numerical ranges of the more general case of nonnormal dual Toeplitz operators with harmonic symbols. In Section 5, we give heuristic proofs of some results based on the concept of lines of support of the numerical range. In Section 6, we briefly discuss the connection between spectral sets and spectra of dual Toeplitz operators. Section 7 is devoted to the quasinormality of dual Toeplitz operators. For the case of dual Toeplitz operators, we introduce and adumbrate the analog of Halmos' fifth problem on the classification of subnormal Toeplitz operators.

2. Preliminaries

Let 𝐓 be a bounded operator on a Hilbert space ℍ. Denote its spectrum by 𝜎(𝐓). The numerical range is a prototype of the spectrum, and it proves useful whenever information about 𝐓 is needed. It is defined by 𝒲(𝐓)∶={⟨𝐓𝜑,𝜑⟩,𝜑∈ℍ,‖𝜑‖=1}. The main involved features of the numerical range are as follows. 𝒲(𝐓) is always bounded and convex, (Toeplitz-Hausdorff theorem). Its closure 𝒲(𝐓) contains the spectrum 𝜎(𝐓). If 𝒲(𝐓) reduces to the singleton {𝜆}, then 𝐓=𝜆𝐼. 𝒲(𝐓) is a linear function of 𝐓, that is, 𝒲(𝛼𝐓+𝛽)=𝛼𝒲(𝐓)+𝛽,∀𝛼,𝛽∈ℂ; hence we see that 𝜔𝒲(𝒮_𝑓)=𝒲(𝒮_𝜔𝑓),for𝜔∈ℂ, and that 𝜆∈𝒲(𝒮_𝑓) implies that 0∈𝒲(𝒮_𝑓−𝜆). If 𝒲(𝐓) is a subset of the real axis, then 𝐓 must be self-adjoint. If 𝐓 is normal and 𝒲(𝐓) is closed, then the extreme points of 𝒲(𝐓) are eigenvalues. If 𝐓 is normal, the closure of 𝒲(𝐓) is the smallest closed convex set containing the spectrum of 𝐓. For further details on the numerical ranges as well as various applications of this pioneering tool in operator theory, see [7, 25–27].

For 𝑓∈ℒ^∞(𝔻), define the dual Toeplitz operator 𝒮_𝑓 to be the operator on (ℒ²_𝑎)^⟂ given by 𝒮_𝑓∶𝑢∈(ℒ²_𝑎)^⟂↦𝒮_𝑓(𝑢)=𝒬(𝑓𝑢)∈(ℒ²_𝑎)^⟂. (2.1) Here 𝒬=𝐼−𝒫 is the familiar orthogonal projection from ℒ²(𝔻,𝑑𝐴) onto (ℒ²_𝑎)^⟂ and 𝒫 is the orthogonal projection from ℒ²(𝔻,𝑑𝐴) onto the Bergman space ℒ²_𝑎. Moreover, for 𝑓∈ℒ^∞(𝔻) and 𝑔∈ℌ^∞, (the algebra of bounded analytic functions) we have 𝒮_𝑓𝑔=𝒮_𝑔𝒮_𝑓,𝒮_𝑓𝑔=𝒮_𝑓𝒮_𝑔,𝒮_𝑓𝒮_𝑔=𝒮_𝑓𝑔−𝐻_𝑓𝐻^∗_𝑔, (2.2)(2.3) where 𝐻_𝑓∶𝑢∈ℒ²_𝑎↦𝐻_𝑓𝑢=𝒬(𝑓𝑢)∈(ℒ²_𝑎)^⟂ is the Hankel operator. If 𝐸 is a subset of ℂ, then the convex hull of 𝐸, denoted by ℋ(𝐸), is the smallest convex set containing 𝐸. Useful properties of the convex hull are well known [26, 28]. For instance, we know that in general the convex hull of an open set is open and the convex hull of a compact set is compact. The following easy property will be also used

Remark 2.1. If 𝐴 is a bounded subset of a finite dimensional normed space, then we have ℋ(𝐴)=ℋ(𝐴). Indeed, observe that 𝐴⊂𝐴 implies that ℋ(𝐴)⊂ℋ(𝐴), whence ℋ(𝐴)⊂ℋ(𝐴)=ℋ(𝐴). Conversely, since 𝐴⊂ℋ(𝐴), we infer that ℋ(𝐴)⊂ℋ(ℋ(𝐴))=ℋ(𝐴), as ℋ(𝐴) is convex.

The following main spectral properties of 𝒮_𝑓 are due to Stroethoff and Zheng [1].

Theorem 2.2. (1) If 𝑓 is in ℒ^∞(𝔻), then ℛ(𝑓)⊂𝜎_𝑒(𝒮_𝑓)⊂𝜎(𝒮_𝑓)⊂ℋ(ℛ(𝑓)).
(2) Let 𝑓 be a continuous real-valued function on 𝔻, then 𝜎(𝒮_f)=𝜎_𝑒(𝒮_𝑓)=𝑓(𝔻).
(3) If 𝑓 is a bounded analytic or coanalytic function on 𝔻, then 𝜎(𝒮_𝑓)=𝜎_𝑒(𝒮_𝑓)=𝑓(𝔻).

For our purpose, we now prove the following useful facts about dual Toeplitz operators.

Lemma 2.3. Let 𝑓 be in ℒ^∞(𝔻). Then, 𝒮_𝑓 is self-adjoint if and only if 𝑓 is real.

Proof. 𝒮_𝑓 is self-adjoint means that 𝒮_𝑓=𝒮^∗_𝑓. This is equivalent to the fact that 𝑓=𝑓, since ‖𝒮_𝑔‖=‖𝑔‖_∞, for 𝑔∈ℒ^∞(𝔻), which is equivalent to the fact that 𝑓 is real-valued.

Lemma 2.4. Let 𝑓 be in ℒ^∞(𝔻). Then, 𝒮_𝑓≥0 if and only if 𝑓≥0.

Proof. If 𝑓≥0, then ⟨𝒮_𝑓𝑔,𝑔⟩=⟨𝑄(𝑓𝑔),𝑔⟩=⟨𝑓𝑔,𝑔⟩=∫_𝔻𝑓(𝑧)|𝑔(𝑧)|²𝑑𝐴(𝑧)≥0,∀𝑔∈(ℒ²_𝑎)^⟂. Conversely, suppose that 𝒮_𝑓≥0, then in particular its spectrum lies in [0,∞). By part (1) of Theorem 2.2, we obtain ℛ(𝑓)⊂𝜎(𝒮_𝑓)⊂[0,∞), whence 𝑓≥0.

Corollary 2.5. (i) A bounded dual Toeplitz operator with a real spectrum must be self-adjoint.
(ii) A bounded dual Toeplitz operator with spectrum lying in the positive real half-axis must be positive.

Proof. This result follows from part (1) of Theorem 2.2 and Lemmas 2.3 and 2.4.

Theorem 2.6. Let 𝑓 be a nonconstant bounded harmonic real-valued function on 𝔻. Then, the operator 𝒮_𝑓 has no eigenvalues.

Proof. Since in general for a constant 𝜆 we have 𝒮_𝑓−𝜆=𝒮_𝑓−𝜆 and 𝑓−𝜆 is harmonic if 𝑓 is, it suffices to show that 𝒮_𝑓𝑔=0 implies that 𝑔=0,∀𝑔∈(ℒ²_𝑎)^⟂. If 𝒮_𝑓𝑔=𝒬(𝑓𝑔)=0, then 𝑓𝑔∈ℒ²_𝑎. Let ℎ∈ℌ^∞(𝔻), then ℎ𝑔∈(ℒ²_𝑎)^⟂. Indeed, ‖ℎ𝑔‖₂≤‖ℎ‖_∞‖𝑔‖₂, and ⟨ℎ𝑔,𝑢⟩=⟨𝑔,ℎ𝑢⟩=0,∀𝑢∈ℒ²_𝑎, (as ℎ𝑢∈ℒ²_𝑎 and 𝑔∈(ℒ²_𝑎)^⟂). It follows that ⟨𝑓𝑔,ℎ𝑔⟩=0. Taking real parts and noticing that 𝑓 is real, we see that ℛ𝑒(⟨𝑓𝑔,ℎ𝑔⟩)=ℛ𝑒_𝔻𝑓𝑔(ℎ𝑔)𝑑𝐴=_𝔻𝑓|𝑔|²ℛ𝑒(ℎ)𝑑𝐴=0. (2.4) Since ℛ𝑒(ℌ^∞) is weak ∗-dense in the set of bounded real harmonic functions [29], we can replace ℛ𝑒(ℎ) with 𝑓 in the latter to obtain 𝑓²|𝑔|²=0. This implies that 𝑓² and |𝑔|² have disjoint supports. However, the harmonic function 𝑓 cannot vanish on a set of positive measure without being zero, whence 𝑔=0.

3. Characterization of the Numerical Range

An operator 𝐓∶𝐌→𝐌 is said to be subnormal if it admits an extension 𝐒∶𝐄→𝐄, such that 𝐌⊂𝐄, 𝐒 is normal, and 𝐌 is invariant under 𝐒. It is well known that if an operator 𝐓 is subnormal, then it is convexoid, that is, 𝒲(𝐓)=ℋ(𝜎(𝐓)). First, the following observation is worth stressing.

Proposition 3.1. Suppose that 𝑓∈ℌ^∞(𝔻). Then 𝒮_𝑓 and 𝒮_𝑓 are convexoid, that is, 𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)), and 𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)).

Proof. Let 𝑓∈ℌ^∞(𝔻), then the multiplication operator ℳ_𝑓∶ℒ²(𝔻)→ℒ²(𝔻) is a normal extension of 𝒮_𝑓. Indeed, we have ℳ_𝑓ℳ_𝑓=ℳ_𝑓ℳ_𝑓 and ℳ_𝑓((ℒ²_𝑎)^⟂)={𝑓𝑔,𝑔∈(ℒ²_𝑎)^⟂}⊂(ℒ²_𝑎)^⟂. Moreover, it is clear that 𝒮_𝑓 is the restriction of ℳ_𝑓 to (ℒ²_𝑎)^⟂. Thus 𝒮_𝑓 is subnormal, whence 𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)).
Concerning the other part of the Proposition, of course 𝒮_𝑓 is not necessarily subnormal, nevertheless we obtain a similar result by exploring the fact that 𝒲(𝒮_𝑓)={𝜆,𝜆∈𝒲(𝒮_𝑓)} and 𝜎(𝒮_𝑓)={𝜆,𝜆∈𝜎(𝒮_𝑓)}. Hence, we obtain 𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)).

For bounded analytic or coanalytic symbols, the fact that 𝒮_𝑓 is convexoid comes from the subnormality of this operator as Proposition 3.1 asserts. However, the spectral inclusion theorem, (namely, part (1) of Theorem 2.2), refines this result. Indeed, making use of the spectral inclusion property, it turns out that all bounded dual Toeplitz operators are convexoid; this represents the aim of the following assertion.

Proposition 3.2. The closure of the numerical range of a bounded dual Toeplitz operator is the convex hull of its spectrum, that is, 𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)), for 𝑓∈ℒ^∞(𝔻).

Proof. Consider the multiplication operator ℳ_𝑓 on ℒ²(𝔻,𝑑𝐴). It is known to be normal, whence convexoid. Thus 𝒲(ℳ_𝑓)=ℋ(𝜎(ℳ_𝑓)). By Problem 67 of [7], we see that 𝜎(ℳ_𝑓)=ℛ(𝑓). By the spectral inclusion Theorem, we see that 𝜎(ℳ_𝑓)=ℛ(𝑓)⊂𝜎(𝒮_𝑓). This yields the following inclusions 𝒲(ℳ_𝑓)=ℋ(𝜎(ℳ_𝑓))=ℋ(ℛ(𝑓))⊂ℋ(𝜎(𝒮_𝑓)). Now, since ℳ_𝑓 is the minimal normal dilation of 𝒮_𝑓, we see that 𝒲(𝒮_𝑓)⊂𝒲(ℳ_𝑓), which is clear from the definition of the numerical range and the fact that 𝒮_𝑓 is the compression of ℳ_𝑓. Therefore, we obtain the first inclusion 𝒲(𝒮_f)⊂ℋ(𝜎(𝒮_𝑓)).
The reverse inclusion is easier: we have 𝜎(𝒮_𝑓)⊂𝒲(𝒮_𝑓), which implies that ℋ(𝜎(𝒮_𝑓))⊂ℋ(𝒲(𝒮_𝑓))=𝒲(𝒮_𝑓), since 𝒲(𝒮_𝑓) is convex.

Remark 3.3. In connection with Proposition 3.2, we ask whether all elements of the dual Toeplitz algebra 𝒟𝒯, (the C*-algebra generated by {𝒮_𝑓,𝑓∈ℌ^∞}), are convexoid, according to the fact that it is generated by subnormal operators.

Now, we are going to characterize the numerical ranges of dual Toeplitz operators with bounded harmonic symbols. First, we make the following observation.

Remark 3.4. According to part (2) of Theorem 2.2, for a nonconstant bounded continuous real-valued function 𝑓 on 𝔻, we infer that 𝜎(𝒮_𝑓)=ℛ(𝑓)=𝑓(𝔻) is an interval. As 𝑓 is bounded, we deduce that 𝜎(𝒮_𝑓)=[inf𝑓,sup𝑓]. Obviously, we have ℋ(𝜎(𝒮_𝑓))=[inf𝑓,sup𝑓].

Lemma 3.5. Suppose that 𝑓 is a nonconstant bounded harmonic real-valued function on 𝔻. Then one has 𝒲(𝒮_𝑓)=(inf𝑓,sup𝑓).

Proof. By Proposition 3.2 and Remark 3.4, we see that 𝒲(𝒮_𝑓)=[inf𝑓,sup𝑓], (the continuity is redundant in the harmonicity). Since 𝒲(𝒮_𝑓) is a convex set whose closure is [inf𝑓,sup𝑓], 𝒲(𝒮_𝑓) contains all elements of [inf𝑓,sup𝑓] except possibly the extreme points inf𝑓 and sup𝑓. Thus (inf𝑓,sup𝑓)⊂𝒲(𝒮_𝑓).
Now, suppose that either inf𝑓 or sup𝑓 belongs to 𝒲(𝒮_𝑓). Then it is an extreme point of 𝒲(𝒮_𝑓), which in fact must be an eigenvalue of 𝒮_𝑓. However, Theorem 2.6 tells us that such 𝒮_𝑓 has no eigenvalues. Thus, we should have 𝒲(𝒮_𝑓)=(inf𝑓,sup𝑓).

Parallel to Brown and Halmos [6] characterization of normal Hardy space Toeplitz operators as well as Axler and Čučković [30] one of normal Bergman space Toeplitz operators with bounded harmonic symbols, normal dual Toeplitz operators were characterized in [1] as follows: for a bounded measurable function 𝑓 on 𝔻, the dual Toeplitz operator 𝒮_𝑓 is normal if and only if the range of 𝑓 lies on a line. Accordingly, we are able to characterize the numerical range of normal dual Toeplitz operators with bounded harmonic symbols.

Theorem 3.6. Let 𝑓 be a nonconstant bounded harmonic function on 𝔻, and suppose that 𝒮_𝑓 is normal. Then, there are (complex) numbers 𝑎,𝑏 such that 𝜎(𝒮_𝑓) is the closed line segment [𝑎,𝑏] and 𝒲(𝒮_𝑓) is the corresponding open line segment (𝑎,𝑏).

Proof. Taking into account the assumption that 𝒮_𝑓 is normal, we are certain from the existence of (complex) constants 𝛼,𝛽 and a real-valued function 𝑔 such that 𝑓=𝛼𝑔+𝛽 whence 𝒮_𝑓=𝛼𝒮_𝑔+𝛽. From the harmonicity of 𝑓 and the linearity of the Laplacian, we see that 𝑔 must be bounded harmonic and real-valued. Now, Lemma 3.5 asserts, therefore, that 𝜎(𝒮_𝑔)=[𝑚,𝑀] and 𝒲(𝒮_𝑔)=(𝑚,𝑀), where 𝑚=inf𝑔 and 𝑀=sup𝑔. Thus 𝜎(𝒮_𝑓)=[𝑎,𝑏] and 𝒲(𝒮_𝑓)=(𝑎,𝑏), (line segments in ℂ), with 𝑎=𝛼𝑚+𝛽, and 𝑏=𝛼𝑀+𝛽.

4. The Numerical Range of a Nonnormal Dual Toeplitz Operator

Lemma 2.4 has a nice consequence on the self-adjointness of certain dual Toeplitz operators.

Theorem 4.1. Let 𝑓∈ℒ^∞(𝔻) be harmonic. Suppose that 𝒲(𝒮_𝑓) lies in the complex upper half-plane and contains some real number. Then 𝒮_𝑓 must be self-adjoint.

Proof. The numerical range lies in the upper half-plane means that ℑ𝑚(⟨𝒮_𝑓𝑔,𝑔⟩)≥0, for ‖𝑔‖=1. Taking 𝑔/‖𝑔‖₂ if necessary, we may conclude that ℑ𝑚(⟨𝒮_𝑓𝑔,𝑔⟩)≥0,∀𝑔∈(ℒ²_𝑎)^⟂. Since ⟨𝒮_𝑓𝑔,𝑔⟩=∫_𝔻𝑓|𝑔|²𝑑𝐴,∀𝑔∈(ℒ²_𝑎)^⟂, taking imaginary parts, we obtain ℑ𝑚(⟨𝒮_𝑓𝑔,𝑔⟩)=_𝔻ℑ𝑚(𝑓)|𝑔|²𝑑𝐴=⟨𝒮_{ℑ𝑚(𝑓)}𝑔,𝑔⟩≥0,∀𝑔∈(ℒ²_𝑎)^⟂. (4.1) This happens only if ℑ𝑚(𝑓)≥0, by Lemma 2.4. Now, suppose that a real number 𝑐∈𝒲(𝒮_𝑓), then there exists some ℎ∈(ℒ²_𝑎)^⟂, with ‖ℎ‖=1, such that ⟨𝒮_𝑓ℎ,ℎ⟩=𝑐. Writing 𝑐 in the form 𝑐=𝑐⟨ℎ,ℎ⟩, we obtain ∫_𝔻(𝑓−𝑐)ℎℎ𝑑𝐴=0. Again, taking imaginary parts we obtain _𝔻ℑ𝑚(𝑓)|ℎ|²𝑑𝐴=0. (4.2) As ℑ𝑚(𝑓)≥0, we deduce that ℑ𝑚(𝑓)|ℎ|²=0 on 𝔻. This implies that ℑ𝑚(𝑓) and |ℎ|² have disjoint supports. Since ‖ℎ‖=1, we deduce that ℎ≠0 on 𝔻. Thus supp(ℑ𝑚(𝑓)) has a positive measure, that is, the harmonic function ℑ𝑚(𝑓) must be zero on a set of nonzero measure. It follows that ℑ𝑚(𝑓)=0 on 𝔻, whence 𝒮_𝑓 is self-adjoint by Lemma 2.3.

Regarding the numerical ranges of a certain class of dual Toeplitz operators, we have the following qualitative characterization.

Theorem 4.2. Let 𝑓∈ℒ^∞(𝔻) be harmonic such that 𝒮_𝑓 is nonnormal. Then 𝒲(𝒮_𝑓) is an open convex set.

Proof. We need only to verify that the convex set 𝒲(𝒮_𝑓) is open. To see this, we proceed by contradiction and suppose that it is not open. Hence it intersects its boundary 𝜕𝒲(𝒮_𝑓). Let 𝜆 be one of such points, that is, 𝜆∈𝒲(𝒮_𝑓)∩𝜕𝒲(𝒮_𝑓), which can be rewritten as 0∈𝒲(𝒮_𝑓−𝜆)∩𝜕𝒲(𝒮_𝑓−𝜆). Now, the convexity of 𝒲(𝒮_𝑓−𝜆) and the fact that 0∈𝜕𝒲(𝒮_𝑓−𝜆) enable us to rotate it so that it lies in the upper half-plane. This means that there exists a unimodular complex number 𝜔=𝑒^𝑖𝜃, for some 𝜃∈[0,2𝜋], such that 𝜔𝒲(𝒮_𝑓−𝜆)=𝒲(𝒮_{𝜔𝑓−𝜔𝜆}) lies in the upper half-plane. By Theorem 4.1, 𝒮_{𝜔𝑓−𝜔𝜆} must be self-adjoint. In other words there exists a real-valued function 𝑔 on 𝔻 such that 𝜔𝑓−𝜔𝜆=𝑔. This implies that 𝑓=𝛼𝑔+𝛽, for some constants 𝛼 and 𝛽. So, we infer that 𝒮_𝑓=𝛼𝒮_𝑔+𝛽 is normal, which contradicts the original hypothesis and completes the proof.

Remark 4.3. Note that Theorems 4.1 and 4.2, (as well as their subsequent corollaries), remain still valid if one assumes merely that the symbols have harmonic imaginary parts.

Corollary 4.4. If 𝑓 is a bounded analytic or coanalytic function, then 𝒲(𝒮_𝑓)=ℋ(𝑓(𝔻)).

Proof. If 𝑓 is constant, the fact is trivially satisfied. If 𝑓 is not constant, then 𝒮_𝑓 is not normal (because the range of an analytic or coanalytic function cannot lie on a line). Hence, by Theorem 4.2, 𝒲(𝒮_𝑓) is an open convex set. On the other hand, by the open mapping theorem 𝑓(𝔻) is open whence ℋ(𝑓(𝔻)) is an open convex set too. Now, by Proposition 3.1 and part (3) of Theorem 2.2 as well as Remark 2.1, we see that 𝒲(𝒮_𝑓)=ℋ(𝑓(𝔻))=ℋ(𝑓(𝔻)). Since an open convex planar set is the interior of its closure, we infer that 𝒲(𝒮_𝑓) coincides with the convex hull of 𝑓(𝔻).

5. Aesthetic Consequences Using Lines of Support of 𝒲(𝒮_𝑓)

A line 𝐿 is said to be a line of support of a planar convex set 𝐾 at a boundary point 𝑃∈𝜕𝐾, if 𝑃∈𝐿 and the set 𝐾 is contained in the closure of one of the two half-planes into which 𝐿 cuts the plane. Clearly, every point on the boundary of a planar convex set lies on a line of support. Based on the concept of lines of support of the numerical range 𝒲(𝒮_𝑓), several interesting consequences of Theorem 4.1 can be observed. Note that the underlying idea goes back to Brown and Halmos [6].

Corollary 5.1. Let 𝑓∈ℒ^∞(𝔻) be harmonic. If a line of support of the numerical range 𝒲(𝒮_𝑓) of the dual Toeplitz operator 𝒮_𝑓 contains a point of 𝒲(𝒮_𝑓), then it contains its whole spectrum 𝜎(𝒮_𝑓) (and hence its entire numerical range 𝒲(𝒮_𝑓)).

Corollary 5.2. Let 𝑓∈ℒ^∞(𝔻) be harmonic. If the spectrum 𝜎(𝒮_𝑓) consists of merely a finite number of eigenvalues, then the operator 𝒮_𝑓 is scalar.

Proofs
Proof of Corollary 5.1
With regard to Corollary 5.1, clearly the line of support can be rotated along with the numerical range till the line will be horizontal and the numerical range above it. Then we translate both in such a way that the line of support will be the real axis; (these two operations correspond to a linear function of the form 𝑧→𝑒^i𝜃𝑧+𝜔, for a fixed complex number 𝜔 and a fixed 𝜃∈[0,2𝜋]). Then one can apply Theorem 4.1 to conclude that 𝑒^𝑖𝜃𝒮_𝑓+𝜔 is self-adjoint, whence 𝑒^𝑖𝜃𝜎(𝒮_𝑓)+𝜔⊂ℝ, and therefore 𝜎(𝒮_𝑓) lies on the original line of support. Since 𝒲(𝒮_𝑓)⊂𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)), by Proposition 3.2, we infer that 𝒲(𝒮_𝑓) lies on the relevant line of support. Proof of Corollary 5.2
Concerning Corollary 5.2, if 𝜎(𝒮_𝑓) consists of a finite number of eigenvalues, then ℋ(𝜎(𝒮_𝑓)) is a polygon with vertices of some of such values. Thus, one can find at least a line of support passing through one of such vertices, which must be in fact an extreme point of 𝒲(𝒮_𝑓). Hence, making use of Corollary 5.1, we infer that all those eigenvalues of 𝒮_𝑓 lie on such line. Consequently, the numerical range is a segment on this line with two extreme points. Taking the line of support perpendicular to the above line, and passing by an endpoint of the segment, and repeating the same procedure, we infer that the numerical range lies on the new line of support as well. A segment lying in two perpendicular lines must be a single point. So, 𝜎(𝒮_𝑓) must be a singleton. By Proposition 3.2 and the properties of the numerical range, we infer that 𝒮_𝑓 must be scalar.

Remark 5.3. Suppose that 𝒮_𝑓 is compact. We know that its spectrum consists of at most a countable number of eigenvalues, including 0. Arguing in the same manner as in the proof of Corollary 5.2, we can show that there are no nonzero compact dual Toeplitz operators with bounded harmonic symbols, which is a particular case of [1, Theorem 7.5].

6. Connection with Spectral Sets

A compact subset 𝔼⊂ℂ containing the spectrum 𝜎(𝑇) of a bounded linear operator 𝑇 acting on a given Hilbert space is called a 𝑘-spectral set for 𝑇 if‖ℱ(𝑇)‖≤𝑘‖ℱ‖_𝔼 (6.1) holds for every function ℱ(𝑧) analytic in a neighborhood of 𝔼 (in [13], rational functions are used instead), where the first norm in the above inequality is the operator norm and the second one is the sup norm over 𝔼. In particular, if 𝑘=1, 1-spectral sets are simply called spectral sets. For more details on spectral sets, we refer to [13–15, 31] and the references therein. For instance, the spectrum of any subnormal operator is a spectral set. The closed unit disk is a spectral set for any contraction.

Our main concern in this section is to describe the dual Toeplitz operators analog of an interesting connection, pointed out by Schreiber [13], between spectral sets and numerical ranges of Toeplitz operators. Let us start with some trivial situations, where 𝜎(𝒮_𝜑) is spectral(1)If 𝒮_𝜑 is a bounded normal dual Toeplitz operator, (i.e., 𝜑(𝔻) lies on a line in ℂ [1]), then 𝜎(𝒮_𝜑) is a spectral set. (2)If 𝜑 is a bounded coanalytic function on the unit disk 𝔻, then 𝜎(𝒮_𝜑) is a spectral set, (since it is subnormal by the proof of Proposition 3.1). (3)If the spectrum of a bounded dual Toeplitz operator 𝒮_𝜑 is a disk, then 𝜎(𝒮_𝜑) is a spectral set too.

However, we can observe that the spectra of dual Toeplitz operators with analytic symbols are also spectral sets for their corresponding operators. This follows from the following observation. If 𝔼⊂ℂ is a planar subset, set 𝔼^∗={𝜔,𝜔∈𝔼}, in particular it can be easily verified that 𝜎(𝑇^∗)=𝜎(𝑇)^∗. Also we adopt the notation 𝑓^∗(𝑧)=𝑓(𝑧) which is an analytic function on 𝔼^∗ whenever 𝑓 is analytic on 𝔼. Then we have the following.

Lemma 6.1. Let 𝑇 be a bounded linear operator on a given Hilbert space. Then, 𝜎(𝑇) is a 𝑘-spectral set for 𝑇 if and only if 𝜎(𝑇)^∗ is a 𝑘-spectral set for its adjoint 𝑇^∗.

Proof. The fact that 𝜎(𝑇) is a 𝑘-spectral set of 𝑇 means that Inequality (6.1) holds, for any ℱ holomorphic in a neighborhood of 𝜎(𝑇). Notice that from the definition of the holomorphic functional calculus we have ‖ℱ^∗(𝑇^∗)‖=‖(ℱ(𝑇))^∗‖=‖ℱ(𝑇)‖. (6.2) On the other hand, we have ‖ℱ^∗‖_𝜎(𝑇)^∗=sup{|ℱ^∗(𝑤)|,𝑤∈𝜎(𝑇)^∗}=sup{|ℱ(𝑧)|,𝑧∈𝜎(𝑇)}=‖ℱ‖_𝜎(𝑇). (6.3) Combining Inequality (6.1) and the last two identities, we infer that ‖ℱ^∗(𝑇^∗)‖≤𝑘‖ℱ^∗‖_𝜎(𝑇^∗₎. (6.4) In order to establish the equivalence, it suffices to observe that if ℱ is holomorphic in a neighborhood of 𝜎(𝑇), then ℱ^∗ is holomorphic in the conjugate of the same neighborhood which contains 𝜎(𝑇)^∗ and conversely.

Therefore, from the above discussion, we deduce the following.

Corollary 6.2. If 𝜑 is a bounded analytic function on the unit disk 𝔻, then 𝜎(𝒮_𝜑) is a spectral set for 𝒮_𝜑.

Similar results for coanalytic Toeplitz operators on both Hardy and Bergman spaces can also be inferred.

Corollary 6.3. (i) If 𝜑∈ℌ^∞(𝜕𝔻), then 𝜎(𝑇_𝜑) is a spectral set for 𝑇_𝜑 defined on ℌ²(𝜕𝔻).
(ii) If 𝜑 is a bounded coanalytic function on the unit disk 𝔻, then 𝜎(𝑇_𝜑) is a spectral set for 𝑇_𝜑 defined on ℒ²_𝑎(𝔻).

7. Some Thoughts on Quasinormal Dual Toeplitz Operators

The Bergman space ℒ²_𝑎 has normalized reproducing kernel 𝑘_𝑤 given by 𝑘_𝑤(𝑧)=1−|𝑤|²(1−𝑤𝑧)². (7.1) Recall that for 𝑤∈𝔻, the involutive disk automorphism 𝜑_𝑤 is defined by 𝜑_𝑤(𝑧)=𝑤−𝑧1−𝑤𝑧,for𝑧∈𝔻. (7.2) For a linear operator 𝐓 on (ℒ²_𝑎)^⟂ and 𝑤∈𝔻, define the operator 𝕊_𝑤(𝐓)∶=𝐓−𝒮_𝜑𝑤𝐓𝓢_𝜑𝑤. A second application of it gives𝕊²_𝑤(𝑇)=𝐓−2𝓢_𝜑𝑤𝐓𝓢_𝜑𝑤+𝓢²_𝜑𝑤𝐓𝓢²_𝜑𝑤. (7.3) For 𝑓,𝑔∈ℒ²(𝔻,𝑑𝐴), define the rank one operator (𝑓⊗𝑔) by (𝑓⊗𝑔)∶ℎ∈ℒ²(𝐷,𝑑𝐴)→(𝑓⊗𝑔)(ℎ)=⟨ℎ,𝑔⟩𝑓∈ℒ²(𝐷,𝑑𝐴). (7.4)

If 𝐓₁ and 𝐓₂ are bounded linear operators on ℒ²(𝔻,𝑑𝐴), then for 𝑓, 𝑔∈ℒ²(𝔻,𝑑𝐴), we have𝐓₁(𝑓⊗𝑔)𝐓^∗₂=(𝐓₁𝑓)⊗(𝐓₂𝑔). (7.5) In the sequel, we will need a formula relating the image of the product 𝐻_𝑓𝐻^∗_𝑔 under the action of the operator 𝕊²_𝑤 and the functions 𝐻_𝑓(𝑘_𝑤) and 𝐻_𝑔(𝑘_𝑤). For 𝑓,𝑔∈ℒ^∞(𝔻), combining (2.3), (7.3), and (7.5), (for a detailed proof see [1, 2]), we obtain𝕊²_𝑤(𝐻_𝑓𝐻^∗_𝑔)=𝐻_𝑓(𝑘_𝑤⊗𝑘_𝑤)𝐻^∗_𝑔=(𝐻_𝑓(𝑘_𝑤))⊗(𝐻_𝑔(𝑘_𝑤)). (7.6) An operator 𝐓 on a Hilbert space is called quasinormal if it commutes with 𝐓^∗𝐓. It is well known that quasinormal operators are subnormal. In what follows we are going to show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols that are not normal.

Theorem 7.1. Let 𝑓 be in ℌ^∞(𝔻), and suppose that 𝒮_𝑓 is quasinormal. Then, the symbol 𝑓 must be constant.

Proof. If 𝑓=0, the conclusion is obvious. So, suppose that 𝑓≠0 and that 𝒮_𝑓 is quasinormal, then we have 𝒮_𝑓𝒮^∗_𝑓𝒮_𝑓=𝒮^∗_𝑓𝒮_𝑓𝒮_𝑓. (7.7) Since 𝑓 is analytic, using Relations (2.2), we obtain 𝒮_𝑓𝒮^∗_𝑓𝒮_𝑓=𝒮_|𝑓|²𝒮_𝑓=𝒮_|𝑓|²_𝑓−𝐻_|𝑓|²𝐻^∗_𝑓,𝒮^∗_𝑓𝒮_𝑓𝒮_𝑓=𝒮_𝑓𝒮_𝑓²=𝒮_𝑓𝑓²−𝐻_𝑓𝐻^∗_𝑓2. (7.8)(7.9) Now, (7.7)–(7.9) yield 𝐻_|𝑓|²𝐻^∗_𝑓=𝐻_𝑓𝐻^∗_𝑓2. (7.10) Introducing the operator 𝕊_𝜔, by (7.6), we see that 𝕊²_𝜔(𝐻_|𝑓|²𝐻^∗_𝑓)=𝐻_|𝑓|²𝑘_𝜔⊗𝐻_𝑓𝑘_𝜔,𝕊²_𝜔(𝐻_𝑓𝐻^∗_𝑓2)=𝐻_𝑓𝑘_𝜔⊗𝐻_𝑓²𝑘_𝜔. (7.11) Combining the latter with (7.10), we obtain 𝐻_|𝑓|²𝑘_𝜔⊗𝐻_𝑓𝑘_𝜔=𝐻_𝑓𝑘_𝜔⊗𝐻_𝑓²𝑘_𝜔. Since 𝑘₀=1, taking 𝜔=0, we obtain 𝐻_|𝑓|²1⊗𝐻_𝑓1=𝐻_𝑓1⊗𝐻_𝑓²1. In other words, by the definition of the rank one operator, we have ⟨𝑢,𝐻_𝑓1⟩𝐻_|𝑓|²1=⟨𝑢,𝐻_𝑓²1⟩𝐻_𝑓1,∀𝑢∈(ℒ²_𝑎)^⟂. (7.12) If 𝐻_𝑓1=0, then 𝑓∈ℒ²_𝑎. Thus 𝑓 must be constant, whence the result follows immediately. If 𝐻_𝑓1≠0, we distinguish several cases as follows. (i)The case 𝐻_|𝑓|²1=0 and 𝐻_𝑓²1=0 cannot happen (because 𝐻_𝑓1=𝒬(𝑓)=𝑓−𝑓(0) and 𝐻_𝑓²1=𝒬(𝑓²)=𝑓²−𝑓²(0) vanish simultaneously). (ii)The similar case 𝐻_|𝑓|²1≠0 and 𝐻_𝑓²1=0 is impossible too, for the same reason. (iii)The case 𝐻_|𝑓|²1=0 and 𝐻_𝑓²1≠0 is impossible too, otherwise ⟨𝑢,𝐻_𝑓²1⟩=0,for all 𝑢∈(ℒ²_𝑎)^⟂; thus 𝐻_𝑓²1∈ℒ²_𝑎 too, whence 𝐻_𝑓²1=0 contradicting the assumption. (iv)If 𝐻_|𝑓|²1≠0 and 𝐻_𝑓²1≠0, then clearly from (7.12) there exists some nonzero constant 𝜆∈ℂ, such that 𝐻_𝑓1=𝜆𝐻_|𝑓|²1,𝐻_𝑓1=𝜆𝐻_𝑓²1. (7.13) Rephrasing (7.13), we see that 𝒬(𝑓−𝜆|𝑓|²)=0 and 𝒬(𝑓−𝜆𝑓²)=0. Thus, 𝑓−𝜆|𝑓|² and 𝑓−𝜆𝑓² are in ℒ²_𝑎, (they are analytic in particular). But, since 𝑓 is analytic, we see that 𝑓(1−𝜆𝑓)=𝑓−𝜆𝑓² is analytic too, whence it is constant. So, set 𝑓(1−𝜆𝑓)=𝜇, for some nonzero complex constant 𝜇. On the other hand, since 𝑓−𝜆|𝑓|²=𝑓(1−𝜆𝑓) is analytic, multiplying by the analytic function 𝑓², we obtain an analytic function, namely, 𝑓(1−𝜆𝑓)𝑓²=|𝑓|²(1−𝜆𝑓)𝑓=𝜇|𝑓|². Now, the function 𝜇|𝑓|², (whose range lies on a line, as |𝑓|² is real-valued and 𝜇 is constant), can be analytic only if it is constant; whence we infer that |𝑓|² is constant. Finally, it is well known that an analytic function with a constant modulus must be constant, whence 𝑓 must be constant.

For bounded conjugate analytic symbols the matter is much more simpler and it uses the Brown-Halmos type Theorem (namely, [1, Theorem 3.1]). Indeed, we have the following.

Theorem 7.2. Let 𝑓 be in ℌ^∞(𝔻), and suppose that 𝒮_𝑓 is quasinormal. Then, 𝑓 must be constant.

Proof. If 𝑓=0, the matter is obvious; so suppose that 𝑓≠0. Since 𝑓 is coanalytic, using Relations (2.2) and (2.3), we obtain 𝒮_𝑓𝒮^∗_𝑓𝒮_𝑓=𝒮_𝑓𝒮_|𝑓|²,𝒮^∗_𝑓𝒮_𝑓𝒮_𝑓=𝒮_|𝑓|²𝒮_𝑓=𝒮_𝑓|𝑓|². (7.14)(7.15) Suppose that 𝒮_𝑓 is quasinormal, then from (7.7), (7.14), and (7.15), we see that 𝒮_𝑓𝒮_|𝑓|²=𝒮_𝑓|𝑓|². Hence 𝒮_𝑓𝒮_|𝑓|² is a dual Toeplitz operator. By the Brown-Halmos type Theorem ([1, Theorem 3.1]), we infer that either 𝑓 is analytic or |𝑓|² is coanalytic. If 𝑓 is analytic, then it is constant since it is coanalytic by hypothesis. If the real function |𝑓|² is coanalytic, then it is constant; whence 𝑓 is constant as well, (as a coanalytic function with constant modulus). Thus, in all cases, we infer that 𝑓 is constant.

Remark 7.3. Clearly if 𝑓 is constant, then 𝒮_𝑓 is normal and then it is quasinormal. So, Theorems 7.1 and 7.2 can be expressed as follows. Let f be a bounded analytic, (or coanalytic), function. Then, 𝒮_f is quasinormal if and only if f is constant, that is, there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols that are not normal.

Corollary 7.4. Let 𝑓 be a bounded analytic or coanalytic function, and suppose that 𝒮_𝑓 is quasinormal. Then, the numerical range of 𝒮_𝑓 reduces to a singleton, that is, 𝒲(𝒮_𝑓)={𝜆} for some complex constant 𝜆.

Proof. Just observe that from Proposition 3.1 one has 𝒲(𝒮_𝑓)=ℋ(𝜎(𝒮_𝑓)). If 𝒮_𝑓 is quasinormal, by Theorems 7.1 and 7.2, we infer that 𝑓=𝜆 for some complex constant 𝜆. The convex hull of a singleton is the set itself. Hence, we obtain 𝒲(𝒮_𝑓)=ℋ({𝜆})={𝜆}.

It seems to be of interest to consider also the typical case of bounded harmonic symbols. It is well known that such functions can be decomposed as 𝑓=𝑓₁+𝑓₂, with Bloch functions 𝑓₁ and 𝑓₂. Here, we confine ourselves to the related case, where 𝑓=𝑔+𝜆𝑔, for 𝑔∈ℌ^∞(𝔻) and 𝜆 a complex constant.

Proposition 7.5. Suppose that 𝑓=𝑔+𝜆𝑔,0≠𝜆∈ℂ, and 0≠𝑔∈ℌ^∞(𝔻). If 𝒮_𝑓 is quasinormal, then 𝒮_𝑓 is normal and 𝜆 must be unimodular.

Proof. Since 𝒮_𝑓 is quasinormal, (7.7) holds. Taking adjoints, we obtain 𝒮_𝑓(𝒮_𝑓𝒮_𝑓−𝒮_𝑓𝒮_𝑓)=0,(𝒮_𝑓𝒮_𝑓−𝒮_𝑓𝒮_𝑓)𝒮_𝑓=0. (7.16) Therefore, using the Hilbert space orthogonality relations, we see that ker([𝒮_𝑓,𝒮_𝑓])=ℛ𝑎𝑛(𝒮_𝑓), which must be nontrivial. Next, a couple of manipulations lead to [𝒮_𝑓,𝒮_𝑓]=(1−|𝜆|²)(𝒮_𝑔𝒮_𝑔−𝒮_𝑔𝒮_𝑔)=(1−|𝜆|²)[𝒮_𝑔,𝒮_𝑔]. (7.17) Now, (2.3) as well as the fact that 𝑔 is analytic yields [𝒮_𝑔,𝒮_𝑔]=−𝐻_𝑔𝐻^∗_𝑔, whence [𝒮_𝑓,𝒮_𝑓]=(|𝜆|²−1)𝐻_𝑔𝐻^∗_𝑔. (7.18) Since the Hankel operator 𝐻_𝑔 is one-to-one (as 𝑔∈ℌ^∞), we infer that 𝐻_𝑔𝐻^∗_𝑔 has a trivial kernel. This contradicts the fact that [𝒮_𝑓,𝒮_𝑓] has a nontrivial kernel, unless [𝒮_𝑓,𝒮_𝑓]=0, which happens only if |𝜆|²=1. Thus 𝒮_𝑓 is normal and 𝜆 must be unimodular.

Corollary 7.6. Let 𝑓 be as in Proposition 7.5 and suppose that 𝒮_𝑓 is quasinormal. Then, 𝒲(𝒮_𝑓)=(𝑎,𝑏) for some complex constants 𝑎 and 𝑏.

Proof. Combining Proposition 7.5 and Theorem 3.6, the result follows.

At this stage, we would like to conclude with a crucial point, which probably sheds some light on the fifth Halmos' problem [32]. This problem asks whether every subnormal Hardy space Toeplitz operator is either normal or analytic. In the Hardy space setting, the original general problem was weakened to whether every quasinormal Toeplitz operator is either normal or analytic, and it was completely solved positively by Amemiya et al. [17], whereas Cowen and Long [19] answered the original problem in the negative. For further results in this direction, see [16, 18, 20–23]. The Bergman space analog seems to be still pending [24]. However, for dual Toeplitz operators, a similar conjecture can be stated with slight modifications, namely: every quasinormal dual Toeplitz operator must be normal. Theorems 7.1 and 7.2, Proposition 7.5 as well as Remark 7.3 support this conjecture.

Acknowledgments

The author would like to express his sincere thanks to the referees for their valuable comments and suggestions. This paper was supported by College of Science-Research Center Project no. (Math/2007/27).