Extremal Domains for Self-Commutators in the Bergman Space

Abstract

In arXiv:1305.5193v1, the authors have shown that Putnam's inequality for the norm of self-commutators can be improved by a factor of $\frac{1}{2}$ for Toeplitz operators with analytic symbol $\phi$ acting on the Bergman space $A^2(\Omega )$. This improved upper bound is sharp when $\phi (\Omega )$ is a disk. In this paper we show that disks are the only domains for which the upper bound is attained.

10 pages