Karel Stroethoff

Toeplitz and Hankel operators on Bergman spaces

In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in C n whose symbols are bounded measurable functions. We giv e necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Topelitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of H ∞(Ω); for these symbols we describe when the Toeplitz or Hankel operators are compact

Generalized Schwarz-Pick estimates

We obtain higher derivative generalizations of the Schwarz-Pick inequality for analytic self-maps of the unit disk as a consequence of recent characterizations of boundedness and compactness of weighted composition operators between Bloch-type spaces.

The Berezin transform and operators on spaces of analytic functions

In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6. Sarason’s example 1. Spaces of analytic functions. In this section we will introduce the spaces of analytic functions on which we will be working. We start with the following general definition. Definition 1.1. A reproducing functional Hilbert space on an open subset Ω of C is a Hilbert space H of functions on Ω such that for every w ∈ Ω the linear functional f 7→ f(w) is bounded on H. If H is a reproducing functional Hilbert space on set Ω, then by the Riesz Representation Theorem for every w ∈ Ω there is a unique element Kw ∈ H for which f(w) = 〈f,Kw〉, for all f ∈ H. We call the function Kw the reproducing kernel at w. Before we turn to a few examples we will prove some simple results about these reproducing kernels. The following proposition gives a way to compute the reproducing kernels. 1991 Mathematics Subject Classification: Primary 47B07, 47B35; Secondary 30C40, 31A05. Research of the author supported by summer grants from the University of Montana and the Montana University system. The paper is in final form and no version of it will be published elsewhere. [361]

Algebraic and spectral properties of dual Toeplitz operators

Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.

Compact Toeplitz operators on weighted harmonic Bergman spaces

We consider the Bergman spaces consisting of harmonic functions on the unit ball in R n that are squareintegrable with respect to radial weights. We will describe compactness for certain classes of Toeplitz operators on these harmonic Bergman spaces.

Schwarz–Pick Type Estimates

We obtain estimates on higher-order derivatives that generalize the classical Schwarz–Pick inequality for analytic self-maps of the unit disk. We discuss further extensions involving certain weight functions, converse results, and versions of all of these results in the setting of the unit ball in .

Nevanlinna-type characterizations for the Bloch space and related spaces

We give a characterisation of the Bloch space in terms of an area version of the Nevanlinna characteristic, analogous to Baernsteins description of the space BMOA in terms of the usual Nevanlinna characteristic. We prove analogous results for the little Bloch space and the space VMOA, and give value distribution characterizations for all these spaces. Finally we give valence conditions on a Bloch or little Bloch function for containment in BMOA or VMOA.

Compact Hankel operators on weighted harmonic Bergman spaces

We prove the compactness of certain Hankel operators on weighted Bergman spaces of harmonic functions on the unit ball in R n .