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 ⊆ C such that the evaluation functionals ϕτ (f ) = f (τ ), τ ∈ , are continuous on H.The Berezin number of an operator X is defined by ber(X) = supτ∈X(τ )= supτ∈Xˆkτ , ˆkτ ,where the operator X acts on the reproducing kernel Hilbert space H = H() over some(non-empty) set . In this paper, we introduce a new family involving means · σt betweenthe Berezin radius and the Berezin norm. Among other results, it is shown that if X ∈ L(H)and f , g are two non-negative continuous functions defined on [0,∞) such that f (t)g(t) =t, (t 0), thenX2σ ber14(f 4(|X|) +g4(|X∗|)) + 12|X|2andX2σ 12ber f 4(|X|) +g2(|X|2) ber f 2(|X|2)+g4(|X∗|),where σ is a mean dominated by the arithmetic mean ∇.