Davis-Wielandt-Berezin radius inequalities of Reproducing kernel Hilbert space operators

TL;DR

This paper establishes bounds for the Davis-Wielandt-Berezin radius of operators on reproducing kernel Hilbert spaces and introduces an inequality relating it to the Berezin number for operator sums.

Contribution

It provides new upper and lower bounds for the Davis-Wielandt-Berezin radius and a novel inequality connecting it with the Berezin number for sums of operators.

Findings

Derived bounds for the Davis-Wielandt-Berezin radius.

Established an inequality involving the Berezin number and the radius.

Extended the understanding of operator behavior in reproducing kernel Hilbert spaces.

Abstract

Several upper and lower bounds of the Davis-Wielandt-Berezin radius of bounded linear operators defined on a reproducing kernel Hilbert space are given. Further, an inequality involving the Berezin number and the Davis-Wielandt-Berezin radius for the sum of two bounded linear operators is obtained, namely, if $A $ and $B$ are reproducing kernel Hilbert space operators, then $$\eta(A+B) \leq \eta(A)+\eta(B)+\textbf{ber}(A^*B+B^*A),$$ where $\eta(\cdot)$ and $\textbf{ber}(\cdot)$ are the Davis-Wielandt-Berezin radius and the Berezin number, respectively.