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Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences

Kevin Esmeral, Egor A. Maximenko

Abstract

In this paper we show that the C*-algebra generated by radial Toeplitz operators with $L_{\infty}$-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric $ρ(j,k)=|\sqrt{\vphantom{jk}j}-\sqrt{\vphantom{jk}k}\,|$. More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.

Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences

Abstract

In this paper we show that the C*-algebra generated by radial Toeplitz operators with -symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric . More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.

Paper Structure

This paper contains 8 sections, 24 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Radial Toeplitz operators
  3. Sqrt-oscillating sequences
  4. Sqrt-oscillating property of the eigenvalues' sequences
  5. Approximation of uniformly continuous functions by convolutions
  6. Approximation of the eigenvalues' sequences by convolutions
  7. Density of $\mathfrak{G}$ in $\operatorname{RO}(\mathbb{Z}_+)$
  8. Beyond the class of bounded generating symbols

Key Result

Lemma 2.4

A function $\varphi\in\,L_{\infty}(\mathbb{C})$ is radial if and only if $\varphi(z)=\mathrm{rad}(\varphi)(z)$ a.e. $z\in\mathbb{C}$.

Figures (2)

  • Figure 1: Scheme of the proof of density: the upper chain represents the approximation of $\sigma(j)$ for large values of $j$ ($j>N$), and the lower one corresponds to the uniform approximation of the sequence $\sigma-\gamma_a$ multiplied by the characteristic function $\chi_{[0,N]}$.
  • Figure 2: The first $301$ values of the sequence $\gamma_a$ from Example \ref{['typicalexample']}.

Theorems & Definitions (52)

  • Definition 2.1: radial operator acting on the Fock space
  • Definition 2.2: radial function
  • Definition 2.3: the radialization of a function
  • Lemma 2.4: criterion for a function to be radial
  • proof
  • Theorem 2.5: criterion of radial operators
  • Proposition 2.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 42 more