Quasihomogeneous Toeplitz operators on the harmonic Bergman space (original) (raw)

Two Questions on Products of Toeplitz Operators on the Bergman Space

2009

The zero product problem and the commuting problem for Toeplitz operators on the Bergman space over the unit disk are some of the most interesting unsolved problems. For bounded harmonic symbols these are solved but for general bounded symbols it is still far from being complete. This paper shows that the zero product problem holds for a special case where one of the symbols has certain polar decomposition and the other is a general bounded symbol. We also prove that the commutant of Tz+z is sum of powers of itself.

Products of Toeplitz Operators on the Bergman Space

Integral Equations and Operator Theory, 2005

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern andCucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).

Characterizations of Bergman space Toeplitz operators with harmonic symbols

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

It is well-known that Toeplitz operators on the Hardy space of the unit disc are characterized by the equality S * 1 T S 1 = T , where S 1 is the Hardy shift operator. In this paper we give a generalized equality of this type which characterizes Toeplitz operators with harmonic symbols in a class of standard weighted Bergman spaces of the unit disc containing the Hardy space and the unweighted Bergman space. The operators satisfying this equality are also naturally described using a slightly extended form of the Sz.-Nagy-Foias functional calculus for contractions. This leads us to consider Toeplitz operators as integrals of naturally associated positive operator measures in order to take properties of balayage into account.

H-Toeplitz operators on the Bergman space

2021

As an extension to the study of Toeplitz operators on the Bergman space, the notion of H-Toeplitz operators Bφ is introduced and studied. Necessary and sufficient conditions under which H-Toeplitz operators become co-isometry and partial isometry are obtained. Some of the invariant subspaces and kernels of H-Toeplitz operators are studied. We have obtained the conditions for the compactness and Fredholmness for H-Toeplitz operators. In particular, it has been shown that a non-zero HToeplitz operator can not be a Fredholm operator on the Bergman space. Moreover, we have also discussed the necessary and sufficient conditions for commutativity of H-Toeplitz operators.

A generalization of Toeplitz operators on the Bergman space

Journal of Operator Theory, 2015

If µ is a finite measure on the unit disc and k ≥ 0 is an integer, we study a generalization derived from Englis's work, T (k) µ , of the traditional Toeplitz operators on the Bergman space A 2 , which are the case k = 0. Among other things, we prove that when µ ≥ 0, these operators are bounded if and only if µ is a Carleson measure, and we obtain some estimates for their norms.

Toeplitz Operators on Generalized Bergman Spaces

Integral Equations and Operator Theory, 2010

We consider the weighted Bergman spaces HL 2 (B d , µ λ), where dµ λ (z) = c λ (1 − |z| 2) λ dτ (z), τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert-Schmidt operators on the generalized Bergman spaces.

Commuting Toeplitz operators with harmonic symbols

Integral Equations and Operator Theory, 1991

This paper shows that on the Bergman space, two Toeplitz operators with harmonic symbols commute only in the obvious cases. The main tool is a characterization of harmonic functions by a conformally invariant mean value property.

Toeplitz operators on generalized Bergman

2014

Abstract. We consider the weighted Bergman spaces HL2(Bd, µλ), where we set dµλ(z) = cλ(1−|z|2)λ dτ(z), with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ> d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded oper-ators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.