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 \(\Theta \subseteq \mathbb{C}\) such that the evaluation functionals \(\varphi _{\tau }\left ( f\right ) =f\left ( \tau \right ) \), \(\tau \in \Theta \), are continuous on ℋ. The Berezin number of an operator \(X\) is defined by \(\mathbf{ber}(X)=\underset{\tau \in {\Theta } }{\sup }\big \vert \widetilde{X}(\tau )\big \vert = \underset{\tau \in {\Theta } }{\sup }\big \vert \langle X\hat{k}_{\tau },\hat{k}_{\tau }\rangle \big \vert \), where the operator \(X\) acts on the reproducing kernel Hilbert space \({\mathscr{H}}={\mathscr{H}(}\Theta )\) over some (non-empty) set \(\Theta \). In this paper, we introduce a new family involving means \(\Vert \cdot \Vert _{\sigma _{t}}\) between the Berezin radius and the Berezin norm. Among other results, it is shown that if \(X\in {\mathscr{L}}({\mathscr{H}})\) and \(f\), \(g\) are two non-negative continuous functions defined on \([0,\infty )\) such that \(f(t)g(t) = t,\,(t\geqslant 0)\), then

$$\begin{aligned} \Vert X\Vert ^{2}_{\sigma }\leqslant \textbf{ber}\left (\frac{1}{4}(f^{4}( \vert X\vert )+g^{4}(\vert X^{*}\vert ))+\frac{1}{2}\vert X\vert ^{2} \right ) \end{aligned}$$

and

$$\begin{aligned} \Vert X\Vert ^{2}_{\sigma }\leqslant \frac{1}{2}\sqrt{\textbf{ber} \left (f^{4}(\vert X\vert )+g^{2}(\vert X\vert ^{2})\right ) \textbf{ber}\left (f^{2}(\vert X\vert ^{2})+g^{4}(\vert X^{*}\vert ) \right )}, \end{aligned}$$

where \(\sigma \) is a mean dominated by the arithmetic mean \(\nabla \).