A Lipschitz estimate for Berezin’s operator calculus

Abstract

F. A. Berezin introduced a general “symbol calculus” for linear operators on reproducing kernel Hilbert spaces. For the particular Hilbert space of Gaussian square-integrable entire functions on complex n n -space, C n \mathbf {C}^{n} , we obtain Lipschitz estimates for the Berezin symbols of arbitrary bounded operators. Additional properties of the Berezin symbol and extensions to more general reproducing kernel Hilbert spaces are discussed.