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Toeplitz operators in Bergman space induced by radial measures

Egor A. Maximenko, Carlos G. Pacheco

Abstract

We study radial Carleson--Bergman measures on the unit disk and the corresponding Toeplitz operators acting in the Bergman space. First, we show that such Toeplitz operators are diagonal in the canonical basis, and we compute their eigenvalue sequences and Berezin transforms in terms of the radial component of the measure. Next, considering the average values of radial measures near the boundary, we give a simple characterization of radial Carleson--Bergman measures. Finally, we prove that the eigenvalue sequences of such Toeplitz operators are Lipschitz continuous with respect to the logarithmic distance on natural numbers. As a consequence, we describe the commutative C*-algebra generated by Toeplitz operators induced by radial Carleson--Bergman measures.

Toeplitz operators in Bergman space induced by radial measures

Abstract

We study radial Carleson--Bergman measures on the unit disk and the corresponding Toeplitz operators acting in the Bergman space. First, we show that such Toeplitz operators are diagonal in the canonical basis, and we compute their eigenvalue sequences and Berezin transforms in terms of the radial component of the measure. Next, considering the average values of radial measures near the boundary, we give a simple characterization of radial Carleson--Bergman measures. Finally, we prove that the eigenvalue sequences of such Toeplitz operators are Lipschitz continuous with respect to the logarithmic distance on natural numbers. As a consequence, we describe the commutative C*-algebra generated by Toeplitz operators induced by radial Carleson--Bergman measures.

Paper Structure

This paper contains 11 sections, 24 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Radial measures and radial operators
  3. Radial measures
  4. Canonical basis of Bergman space
  5. Radial operators
  6. Diagonalization of the Toeplitz operator induced by radial measure
  7. The Berezin transform of a radial measure
  8. Using averages and distribution functions
  9. When a radial measure is Carleson--Bergman measure?
  10. Lipschitz property of the spectral sequences
  11. A class of radial complex measures

Key Result

Theorem 2.4

Let $\mu\colon\mathcal{B}_\mathbb{D}\to[0,+\infty)$ be a finite Borel measure. Then the following conditions are equivalent:

Theorems & Definitions (63)

  • Definition 2.1: radial measure
  • Definition 2.2: radial extension of a measure
  • Remark 2.3
  • Theorem 2.4: decomposition of a radial measure
  • proof
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 53 more