Some extensions of Berezin number inequalities on operators

Abstract

In this paper, we establish some upper bounds for Berezin number inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{ccc}0& X,Y& 0\end{array}\right]$, then \begin{align*} \textbf{ber}^{r}(T)\leq 2^{r-2}\left(\textbf{ber}(f^{2r}(|X|)+g^{2r}(|Y^*|))+\textbf{ber}(f^{2r}(|Y|)+g^{2r}(|X^*|))\right)\\ -2^{r-2} \inf_{\|(k_{λ_{1}},k_{λ_{2}})\|=1} η(k_{λ_{1}},k_{λ_{2}}), \end{align*} where $\eta (k_{{\lambda }_1},k_{{\lambda }_2})={(⟨(f^{2r}(|X|)+g^{2r}(|Y^{\ast }|))k_{{\lambda }_2},k_{{\lambda }_2}⟩}^{\frac{1}{2}}-{⟨\left(f^{2r}(|Y|)+g^{2r}(|X^{\ast }|)\right)k_{{\lambda }_1},k_{{\lambda }_1}⟩}^{\frac{1}{2}}{)}^2$, $X,Y$ are bounded linear operators on a Hilbert space $\mathcal{H}=\mathcal{H}(\Omega )$, $r\ge 1$ and $f$, $g$ are nonnegative continuous functions on $[0,\infty )$ satisfying the relation $f(t)g(t)=t(t\in [0,\infty ))$.

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS