The Berezin symbol and multipliers of functional Hilbert spaces

Abstract:

This paper focuses on a multiplicative property of the Berezin symbol $\tilde A$, of a given linear map $A:\mathcal {H} \mapsto \mathcal {H}$, where $\mathcal {H}$ is a functional Hilbert space of analytic functions. We show $\widetilde {AB} = \tilde A\tilde B$ 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.