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This article aims to survey the convexity of the Berezin range for operators acting on reproducing kernel Hilbert spaces. The Berezin range of a bounded operator T acting on a reproducing kernel Hilbert space \(\mathcal {H}\) is the set \(\textit{Ber}(T)\) := \(\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle _{\mathcal {H}} : x \in X\}\) , where \(\hat{k}_{x}\) is the normalized reproducing kernel for \(\mathcal {H}\) at \(x \in X\) . Karaev [18] initiated the study on the geometry of the Berezin range. In this survey, we thoroughly examine the convexity of the Berezin range for composition operators on both the Hardy space and Bergman space.

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Survey on the Convexity of the Berezin Range

  • 会议论文

  • 发表日期：

    2025年

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  • Athul Augustine,
  • P. Shankar

摘要

This article aims to survey the convexity of the Berezin range for operators acting on reproducing kernel Hilbert spaces. The Berezin range of a bounded operator T acting on a reproducing kernel Hilbert space \(\mathcal {H}\) is the set \(\textit{Ber}(T)\) := \(\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle _{\mathcal {H}} : x \in X\}\) , where \(\hat{k}_{x}\) is the normalized reproducing kernel for \(\mathcal {H}\) at \(x \in X\) . Karaev [18] initiated the study on the geometry of the Berezin range. In this survey, we thoroughly examine the convexity of the Berezin range for composition operators on both the Hardy space and Bergman space.