Some extensions of Berezin number inequalities on operators

Abstract

We establish some upper bounds for Berezin number inequalities including inequalities for $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if , then

$${ber}^r(T)\le 2^{r-2}\left(ber(f^{2r}(\left|X\right|)+g^{2r}(\left|Y^{\ast }\right|))+ber(f^{2r}(\left|Y\right|)+g^{2r}(\left|X^{\ast }\right|))\right)-2^{r-2}\underset{\Vert (k_{{\lambda }_1},k_{{\lambda }_2})\Vert =1}{\inf }\eta (k_{{\lambda }_1},k_{{\lambda }_2}),$$

where $X$, $Y$ are bounded linear operators on a Hilbert space $\mathcal{\mathscr{H}}=\mathcal{\mathscr{H}}(\mathrm{\Omega })$, $r\ge 1$, $f$, $g$ are nonnegative continuous functions on $[0,\infty )$ satisfying the relation $f(t)g(t)=t$ $(t\in [0,\infty ))$ and

$$\eta (k_{{\lambda }_1},k_{{\lambda }_2})={\left({⟨(f^{2r}(\left|X\right|)+g^{2r}(\left|Y^{\ast }\right|))k_{{\lambda }_2},k_{{\lambda }_2}⟩}^{\frac{1}{2}}-{⟨(f^{2r}(\left|Y\right|)+g^{2r}(\left|X^{\ast }\right|))k_{{\lambda }_1},k_{{\lambda }_1}⟩}^{\frac{1}{2}}\right)}^2.$$