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Convexity of the Berezin range. (English) Zbl 07523366

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

MSC:

47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
52A10 Convex sets in \(2\) dimensions (including convex curves)
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