Roots of Toeplitz Operators on the Bergman space

TL;DR

This paper extends the understanding of Toeplitz operators on the Bergman space by establishing the existence of p-th roots for a broader class of symbols, thereby advancing the description of their commutants.

Contribution

It proves the existence of p-th roots for a wider class of symbols beyond monomials, including those with logarithmic terms, enriching the theory of Toeplitz operators.

Findings

Existence of p-th roots for symbols with logarithmic components.

Extension of commutant description to broader symbol classes.

Enhanced understanding of Toeplitz operator structure on Bergman space.

Abstract

One of the major questions in the theory of Toeplitz operators on the Bergman space over the unit disk $\mathbb D$ in the complex plane $\mathbb C$ is a complete description of the commutant of a given Toeplitz operator, that is the set of all Toeplitz operators that commute with it. In \cite{l}, the first author obtained a complete description of the commutant of Toeplitz operator $T$ with any quasihomogeneous symbol $\phi(r)e^{ip\theta}, p>0$ in case it has a Toeplitz p-th root $S$ with symbol $\psi(r)e^{i\theta}$, namely, commutant of $T$ is the closure of the linear space generated by powers $S^n$ which are Toeplitz. But the existence of p-th root was known until now only when $\phi(r)=r^m,m \geq 0$. In this paper we will show the existence of p-th roots for a much larger class of symbols, for example, it includes such symbols for which $$\phi(r)=\sum_{i=1}^kr^{a_i}(\ln r)^{b_i},0\leq a_i, b_i for all 1\leq i\leq k .$$