Berezin number inequalities for Hilbert space operators

Abstract

In this paper, by using of the definition Berezin symbol, we show some Berezin number inequalities. Among other inequalities, it is shown that if $A,B,X\in \mathbb{B}(\mathcal{H})$, then $$\mathbf{ber}(AX\pm XA)⩽{\mathbf{ber}}^{\frac{1}{2}}\left(A^{\ast }A+A{A}^{\ast }\right){\mathbf{ber}}^{\frac{1}{2}}\left(X^{\ast }X+X{X}^{\ast }\right)$$ and $${\mathbf{ber}}^2(A^{\ast }XB)⩽\Vert X{\Vert }^2\mathbf{ber}(A^{\ast }A)\mathbf{ber}(B^{\ast }B).$$