The A-Davis-Wielandt Berezin number of semi Hilbert operators with some related inequalities

Abstract

In this article, the concept of the A-Davis-Wielandt Berezin number is introduced for positive operator $A$. Some upper and
lower bounds for the A-Davis-Wielandt Berezin number are proved. Moreover, some inequalities related to the concept of the Davis-Wielandt Berezin number are obtained, which are generalizations of known results. Among them, it is shown that
\begin{align*}
&\be_{dw}^{2}(S)\\&\leq\inf_{\gamma\in \mathbb{C}}\{\left(2||Re(\gamma)Re(S)+Im(\gamma)Im(S)||+||S^{*}S-2Re(\bar{\gamma}S)||\right)^{2}+2||Re(\bar{\gamma}S)||-|\gamma|^{2}+\be^{2}(S-\gamma I)\},
\end{align*}
where $S\in B(\mathscr{H}(\Omega))$. Also, we determined the exact value of the A-Davis--Wielandt Berezin number of some special type of operator matrices.