Abstract
The Berezin range of a bounded operator T acting on a reproducing kernel Hilbert space \(\mathcal {H}\) is the set \(\text {Ber}(T):= \{\langle T\hat{k}_{x},\hat{k}_{x} \rangle _{\mathcal {H}}: x \in X\}\), where \(\hat{k}_{x}\) is the normalized reproducing kernel for \(\mathcal {H}\) at \(x \in X\). In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators A on the reproducing kernel Hilbert space \(\mathcal {H}(\Omega )\), namely we prove that \(f(\textrm{Ber}(\Phi (A)))=\textrm{Ber}(\Phi (f(A)))\), where \(\Phi :\mathcal {B} \left( \mathcal {H}\left( \Omega \right) \right) \mathcal {\rightarrow }\mathcal {B}\left( \mathcal {K(}Q\mathcal {)}\right) \) is a normalized positive linear map.
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Acknowledgements
The first author is supported by the Junior Research Fellowship (09/0239(13298)/2022-EM) of CSIR (Council of Scientific and Industrial Research, India). The second author was supported by the Researchers Supporting Project number(RSPD2023R1056), King Saud University, Riyadh, Saudi Arabia. The third author is supported by the Teachers Association for Research Excellence (TAR/2022/000063) of SERB (Science and Engineering Research Board, India). First author and third author would like to thank Jaydeb Sarkar for the valuable discussions.
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Communicated by Daniel Alpay.
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Augustine, A., Garayev, M. & Shankar, P. Composition Operators, Convexity of Their Berezin Range and Related Questions. Complex Anal. Oper. Theory 17, 126 (2023). https://doi.org/10.1007/s11785-023-01432-x
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DOI: https://doi.org/10.1007/s11785-023-01432-x
Keywords
- Berezin transform
- Berezin range
- Berezin set
- Convexity
- Composition operator
- Hardy space
- Bergman space
- Berezin set mapping theorem