A New Family of Semi-Norms Between the Berezin Radius and the Berezin Norm

Abstract

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set such that the evaluation functionals , , are continuous on ℋ. The Berezin number of an operator is defined by , where the operator acts on the reproducing kernel Hilbert space over some (non-empty) set . In this paper, we introduce a new family involving means between the Berezin radius and the Berezin norm. Among other results, it is shown that if and , are two non-negative continuous functions defined on such that , then

and

where is a mean dominated by the arithmetic mean .