On the commutativity of a certain class of Toeplitz operators

Hashem Alsabi; Issam Louhichi

paper

On the commutativity of a certain class of Toeplitz operators

arXiv:1704.04757

Abstract

In this paper we prove that if the polar decomposition of a symbol $f$ is truncated above, i.e., $f (r e^{i θ}) = \sum_{k = - \infty}^{N} e^{ik θ} f_{k} (r)$ where the $f_{k}$ 's are radial functions, and if the associated Toeplitz operator $T_{f}$ commutes with $T_{z^{2} + z ˉ^{2}}$ , then $T_{f} = Q (T_{z^{2} + z ˉ^{2}})$ 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].