Convexity of the Berezin Range

Abstract

This paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B(T ) := { T kx, kx H : x ∈ X}, where kx is the normalized reproducing kernel for H at x ∈ X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.

Spectra for Toeplitz Operators Associated with a Constrained Subalgebra

“…This transform appears naturally in the study of various operator theoretic properties of Toeplitz operators (e.g. see [11,Section 2]).…”