Toeplitz operators with uniformly continuous symbols
type: Item
instance of: scholarly article
title: Toeplitz operators with uniformly continuous symbols
zbMATH Open document ID: 1351.32030
author: Wolfram Bauer
author: L. A. Coburn
published in: Integral Equations and Operator Theory
publication date: 2015-10-23T00:00:00Z
review text: Toeplitz operators \(T_f\) on Bergman spaces of bounded symmetric domains (BSD) \(\Omega\) in complex space \(\mathbb{C}^n\) or on the Segal-Bargmann space of Gaussian square-integrable complex functions on \(\mathbb{C}^n\) are studied. The symbols are uniformly continuous (UC) functions with respect to the Bergman (hyperbolic) distance function \(\beta(\cdot , \cdot)\) on \(\Omega\) or with respect to the usual Euclidean distance \(d(\cdot , \cdot)\) on \(\mathbb{C}^n\). For \(f\) in UC, the authors prove in Theorem 1.1 that the operator \(T_f\) is bounded iff \(f\) is bounded; \(T_f\) is compact iff \(f\) vanishes at the boundary \(\partial\Omega\) (or at \(\infty\) for \(\mathbb{C}^n\)). The Bergman metric induced by the kernel \(K\) up to a factor coincides with the Euclidean metric \(d(z,w) := |z-w|\) on \(\mathbb{C}^n\). The authors write \(\mathrm{Lip}(\mathbb{C}^n)\) and \(\mathrm{UC}(\mathbb{C}^n)\) for the space of all Lipschitz continuous and uniformly continuous functions with respect to \(d\). Then \(\mathrm{Lip}(\mathbb{C}^n)\subset\mathrm{UC}(\mathbb{C}^n)\). Let \(f \in\mathrm{UC}(\mathbb{C}^n)\). They write \(\mathcal{T}{\mathbb{C}^n}\) for the ``symbol space'' of all functions \(f : \mathbb{C}^n \rightarrow \mathbb{C}\) satisfying \(f(w- \cdot) \in L^2(\mathbb{C}^n, d\mu)\) for all \(w \in \mathbb{C}^n\). For a given function \(f \in \mathcal{T}{\mathbb{C}^n}\) and \(t \in (0, 1/4]\), they define the heat transform of \(f\) at time \(t\) as \[ \tilde{f}^{(t)}(w) := \frac{1}{(4\pi t)^n}\int_{\mathbb{C}^n} f(w-z)e^{-\frac{|z|^2}{4t}} dv(z). \] Then the authors prove in Theorem 2.1 that \(\lim_{t \downarrow 0} \tilde{f}^{(t)} = f\) uniformly on \(\mathbb{C}^n\). The mean oscillation of \(f\) is denoted by \(\mathrm{MO}(f,w)\). The authors write \(\mathrm{BMO}^2(\mathbb{C}^n)\) for the space of all functions of bounded mean oscillation \(f \in \mathcal{T}{\mathbb{C}^n}\) satisfying \(\| f\|_{\mathrm{BMO}} := \sup_{z \in \mathbb{C}^n} \mathrm{MO}(f,z)^{…
reviewed by: Takanori Yamamoto
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 6497570
zbMATH Keywords: bounded symmetric domains
zbMATH Keywords: Segal-Bargmann space
zbMATH Keywords: heat transform
zbMATH Keywords: Bergman metric
MaRDI profile type: Publication
full work available at URL: s00020-015-2235-4
OpenAlex ID: W2018105725
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DOI: 10.1007/S00020-015-2235-4
description: scientific article; zbMATH DE number 6497570