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A New Family of Semi-Norms Between the Berezin Radius and the Berezin Norm
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Journal Article
A New Family of Semi-Norms Between the Berezin Radius and the Berezin Norm
2024
Overview
A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set Θ⊆C such that the evaluation functionals φτ(f)=f(τ), τ∈Θ, are continuous on ℋ. The Berezin number of an operator X is defined by ber(X)=supτ∈Θ|X˜(τ)|=supτ∈Θ|〈Xkˆτ,kˆτ〉|, where the operator X acts on the reproducing kernel Hilbert space H=H(Θ) over some (non-empty) set Θ. In this paper, we introduce a new family involving means ∥⋅∥σt between the 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), then ∥X∥σ2⩽ber(14(f4(|X|)+g4(|X∗|))+12|X|2) and ∥X∥σ2⩽12ber(f4(|X|)+g2(|X|2))ber(f2(|X|2)+g4(|X∗|)), where σ is a mean dominated by the arithmetic mean ∇.
Publisher
Springer Nature B.V
Subject
