THE BEREZIN SYMBOL AND MULTIPLIERS OF FUNCTIONAL HILBERT-SPACES

This paper focuses on a multiplicative property of the Berezin symbol (A) over tilde, of a given linear map A: H --> H, where H is a functio nal Hilbert space of analytic functions. We show (AB) over tilde = (AB ) over tilde for all B in B(H) if and only if A is a multiplication op erator M(phi) where phi is a multiplier. We also present a version of this result for vector-valued functional Hilbert spaces.