Berezin Number and Norm Inequalities for Operators in Hilbert and Semi-Hilbert Spaces

CRISTIAN CONDE; KAIS FEKI ; FUAD KITTANEH

Matrix and Operator Equations and Applications

Springer Cham

Año: 2023; p. 1 - 34

Let $\big(\mathcal{H}_\Omega,\langle \cdot, \cdot \rangle\big)$ be the reproducing kernel Hilbert space over some (non-empty) set $\Omega$. Let $\widehat{k}_{\lambda}$ and $\widehat{k}_{\mu}$ denote two normalized reproducing kernels of $\mathcal{H}_\Omega$. The Berezin number and the Berezin norm of a bounded linear operator $T$ acting on $\mathcal{H}_\Omega$ are, respectively, given by $\ber(T)=\underset{\lambda \in\Omega}{\sup }\big|\langle T\widehat{k}_{\lambda},\widehat{k}_{\lambda}\rangle\big|$ and $\|T\|_{\ber}=\underset{\lambda,\mu \in\Omega}{\sup }\big|\langle T\widehat{k}_{\lambda},\widehat{k}_{\mu}\rangle\big|$. Our aim in this chapter is to present several inequalities involving $\ber(\cdot)$ and $\|\cdot\|_{\ber}$. In addition, some bounds related to the Berezin number and Berezin norm are established when an additional semi-inner product structure induced by a positive operator $A$ on $\mathcal{H}_\Omega$ is considered.