Berezin number inequalities for Hilbert space operators

TL;DR

This paper establishes new inequalities involving the Berezin number for operators on Hilbert spaces, enhancing understanding of operator bounds using Berezin symbols.

Contribution

It introduces novel Berezin number inequalities for Hilbert space operators, expanding the theoretical framework of operator analysis.

Findings

Derived inequalities for operator Berezin numbers.

Established bounds involving operator products and sums.

Provided mathematical tools for operator norm estimations.

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 $A, B, X\in{\mathbb{B}}(\mathscr H)$, then $$\mathbf{ber}(AX\pm XA)\leqslant \mathbf{ber}^{\frac{1}{2}}\left(A^*A+AA^*\right)\mathbf{ber}^{\frac{1}{2}}\left(X^*X+XX^*\right)$$ and $$\mathbf{ber}^2(A^*XB)\leqslant\|X\|^2\mathbf{ber}(A^*A)\mathbf{ber}(B^*B).$$