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Commutant of sum of two quasihomogeneous Toeplitz operators

Aissa Bouhali, Issam Louhichi

Abstract

A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is to fully characterize the set of all Toeplitz operators that commute with a given one. In [2], the second author described the sum $S = T_{e^{imθ}f} +T_{e^{ilθ}g}$, where $f$ and $g$ are radial functions, that commutes with the sum $T = T_{e^{ipθ}r(2M+1)p} +T_{e^{isθ}r(2N+1)s}$ . It is proved that $S = cT$, where $c$ is a constant. In this article, we shall replace $r^{(2M+1)p}$ and $r^{(2N+1)s}$ by $r^n$ and $r^d$ respectively, with $n$ and $d$ in $\mathbb{N}$, and we shall show that the same result holds.

Commutant of sum of two quasihomogeneous Toeplitz operators

Abstract

A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is to fully characterize the set of all Toeplitz operators that commute with a given one. In [2], the second author described the sum , where and are radial functions, that commutes with the sum . It is proved that , where is a constant. In this article, we shall replace and by and respectively, with and in , and we shall show that the same result holds.
Paper Structure (3 sections, 7 theorems, 69 equations)

This paper contains 3 sections, 7 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and tools
  3. Main Results

Key Result

Theorem 1

Let $\phi(r)=r^{(2M+1)p}$ and $\psi(r)=r^{(2N+1)s}$ with $p<s$, $M$, and $N$ being all integers greater or equal to $1$. Assume there exist $m, l\in\mathbb{N}$ and nontrivial radial functions $f, g$ such that the following two hypotheses are satisfied Then $m=p,\ l=s$ and for some constant $c$.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Theorem 3
  • proof
  • Theorem 4
  • proof