An extension of the Euclidean Berezin number

Abstract

The Berezin transform ? of an operator A, acting on the reproducing kernel Hilbert space H = H(?) over some (non-empty) set ?, is defined by ?(?) = ?A?k?,?k?? (? ? ?), where ?k? = k?/?k?? is the normalized reproducing kernel of H. The Berezin number of an operator A is defined by ber(A) = sup ??? ??(?)? = sup ??? ??A?k?,?k???. In this paper, by using the definition of 1-generalized Euclidean Berezin number, we obtain some possible relations and inequalities. It is shown, among other inequalities, that if Ai ? L(H(?)) (i = 1,..., n), then berg(A1,...,An) ? g?1 (n?i=1 g(ber(Ai))) ? n?i=1 ber(Ai), in which g?[0,?) ? [0,?) is a continuous increasing convex function such that g(0) = 0.