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The Berezin symbol and multipliers of functional Hilbert spaces. (English) Zbl 0847.46010

Summary: This paper focuses on a multiplicative property of the Berezin symbol \(\widetilde A\), of a given linear map \(A: {\mathcal H}\mapsto {\mathcal H}\), where \(\mathcal H\) is a functional Hilbert space of analytic functions. We show \(\widetilde{AB}= \widetilde A\widetilde B\) for all \(B\) in \({\mathcal B}({\mathcal H})\) if and only if \(A\) is a multiplication operator \(M_\varphi\), where \(\varphi\) is a multiplier. We also present a version of this result for vector-valued functional Hilbert spaces.

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B38 Linear operators on function spaces (general)
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