Convexity of the Berezin range

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 k̂x, k̂x〉H : x ∈ X}, where k̂x 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.

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