# On the commutativity of a certain class of Toeplitz operators

## TL;DR

This paper characterizes when certain Toeplitz operators with truncated polar decompositions commute with a specific quadratic operator, showing they are essentially linear functions of that operator, thus addressing an open problem.

## Contribution

It proves that Toeplitz operators with truncated polar decompositions commuting with a specific quadratic Toeplitz operator are polynomially related, providing a partial solution to an open problem.

## Key findings

- Toeplitz operators with truncated polar decompositions commute with the quadratic operator only if they are linear functions of it.
- The polynomial relating such Toeplitz operators has degree at most 1.
- The result partially solves an open problem by Axler, Cuckovic, and Rao.

## Abstract

In this paper we prove that if the polar decomposition of a symbol $f$ is truncated above, i.e., $f(re^{i\theta} )=\sum_{k=-\infty}^Ne^{ik\theta} f_k (r)$ where the $f_k$'s are radial functions, and if the associated Toeplitz operator $T_f$ commutes with $T_{z^2+\bar{z}^2}$, then $T_f=Q(T_{z^2+\bar{z}^2})$ where $Q$ is a polynomial of degree at most $1$. This gives a partial answer to an open problem by S. Axler, Z. Cuckovic and N. V. Rao [2, p. 1953].