Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

Abstract

Abstract. Let Ω ⊂ C d be an irreducible bounded symmetric domain of type (r, a, b) in its Harish-Chandra realization. We study Toeplitz operators T ν g with symbol g acting on the standard weighted Bergman space H 2 ν over Ω with weight ν. Under some conditions on the weights ν and ν0 we show that there exists C(ν, ν0) > 0, such that the Berezin transformgν 0 of g with respect to H 2 ν 0 satisfies:g , for all g in a suitable class of symbols containing L ∞ (Ω). As a consequence we apply a result in Engliš (Integr Equ Oper theory 33: 1999), to prove that the compactness of T ν g is independent of the weight ν, whenever g ∈ L ∞ (Ω) and ν > C where C is a constant depending on (r, a, b).Mathematics Subject Classification (2010). 32A36, 32M15, 53C35.

The Essential Norm of Operators on $${A^p_\alpha(\mathbb{B}_n)}$$

“…Moreover, as we will see in this paper, it is possible to obtain a characterization of compact operators on A p α in terms of the Berezin transform. The following papers provide additional examples of how the Berezin transform determines properties of several classes of operators on the Bergman space of the unit ball B n , [2,8,9,11,13,14,17].…”

A reproducing kernel thesis for operators on Bergman-type function spaces

“…Results of this kind are numerous in the literature. Here we mention only a few and refer the reader to [5,13,17,18,21,23,24,29,32,37] and the references in those papers for more examples of these results. The RKT for compactness was first proved to hold for every Toeplitz operator on the classical Bergman space of the disc by Zheng [38].…”

Construction of some classes of commutative Banach and C⋆-algebras of Toeplitz operators

“…In our case, we show that the operators in (5) are unitary equivalent to a composition of a shift and a multiplication operator on L 2 (R n ).…”

“…More precisely, these algebras arise from symbol classes that are radial with respect to geodesics. In the case of the Segal-Bargmann space, as well as of Bergman spaces over general bounded symmetric domains [5], a geometric characterization for obtaining the algebraic property of commutativity is not clear. Motivated by the work done in [14], for the case of the unit ball in higher dimensions, one has to include Banach algebras of commuting Toeplitz operators.…”