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Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Armando Sánchez-Nungaray, José Rosales-Ortega, Carlos González-Flores

Abstract

We consider symmetric separately radial (with corresponding group $S_n\rtimes \mathbb{T}^n$) and alternating separately radial (with corresponding group $A_n\rtimes \mathbb{T}^n$) symbols, as well as the associated Toeplitz operators on the weighted Bergman spaces on the unit ball on $\mathbb{C}^n$. Using a purely representation theoretic approach we obtain that the $C^*$-algebras generated by each family of such Toeplitz operators is commutative. Furthermore, we show that the symmetric separately radial Toeplitz operators are more general than radial Toeplitz operators, i.e., every radial Toeplitz operator is a symmetric separately radial.

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Abstract

We consider symmetric separately radial (with corresponding group ) and alternating separately radial (with corresponding group ) symbols, as well as the associated Toeplitz operators on the weighted Bergman spaces on the unit ball on . Using a purely representation theoretic approach we obtain that the -algebras generated by each family of such Toeplitz operators is commutative. Furthermore, we show that the symmetric separately radial Toeplitz operators are more general than radial Toeplitz operators, i.e., every radial Toeplitz operator is a symmetric separately radial.
Paper Structure (7 sections, 10 theorems, 74 equations, 2 tables)

This paper contains 7 sections, 10 theorems, 74 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $S_n \rtimes \mathbb{T}^n$-intertwining operators
  4. $A_n \rtimes \mathbb{T}^n$-intertwining operators
  5. Toeplitz operators with symmetric separately radial symbols
  6. Toeplitz operators with alternating separately radial symbols
  7. Toeplitz operators with anti-symmetric separately radial symbols

Key Result

Proposition 1

If $a\in L^{\infty}(\mathbb{B}^n)$ satisfies $a\circ ( \sigma, z) = a$, for every $( \sigma, z) \in S_n \rtimes \mathbb{T}^n$, then, for every $\alpha > -1$, the Toeplitz operator $T_a$ is an element of $\hbox{End}_{ S_n\rtimes \mathbb{T}^n } (\mathcal{H}^2_{\alpha}(\mathbb{B}^n))$.

Theorems & Definitions (27)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Proposition 2
  • proof
  • Theorem 3
  • Corollary 1
  • Proposition 4
  • proof
  • ...and 17 more