Toeplitz operators on harmonic Bergman spaces

Abstract

Abstract. We study Toeplitz operators on the harmonic Bergman spaces on bounded smooth domains. Two classes of symbols are considered; one is the class of positive symbols and the other is the class of uniformly continuous symbols. For positive symbols, boundedness, compactness, and membership in the Schatten classes are characterized. For uniformly continuous symbols, the essential spectra are described.

Positive Toeplitz Operators of Schatten–Herz Type

“…In case dµ = ϕ dV , we write T µ = T ϕ . Note that T µ is defined on a dense subset of b 2 , because bounded harmonic functions form a dense subset of b 2 ; see, for example, Lemma 2.5 of [2]. A Toeplitz operator T µ is called positive if µ ∈ M is a positive (finite) Borel measure (we will simply write µ ≥ 0).…”

“…whenever x ∈ B and y ∈ E δ 0 (x); see Lemma 2.3 of [2] on general domains. A consequence useful for our purpose is the fact that averaging functions over balls of small radii are dominated by Berezin transforms.…”

The Atomic Decomposition of Harmonic Bergman Functions, Dualities and Toeplitz Operators

“…We also need a characterization for Schatten p-class Toeplitz operators with positive symbol. For the proof, see [12,Theorem 11] or [4,Theorem 3.13]. In the following, the measure dλ is defined on B by dλ(x) = (1 − |x|) −n dV (x).…”

Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

“…Currently, very little seems to be known about these Toeplitz operators [8,18,23,26,29], and even less about the corresponding Berezin transforms [15].…”