Numerical radius and Berezin number inequality

TL;DR

This paper investigates inequalities involving the numerical radius and Berezin number of operators on Hilbert spaces, providing new bounds, properties, and examples, especially for Toeplitz operators and isometries.

Contribution

It introduces new inequalities and properties for numerical radius and Berezin number, including multiplicativity, sub-multiplicativity, and Hardy-type bounds, with applications to Toeplitz operators.

Findings

Numerical radius of a pure two-isometry is 1.

Crawford number of a pure two-isometry is 0.

Numerical radius of Toeplitz operators with non-constant inner functions is 1.

Abstract

We study various inequalities for numerical radius and Berezin number of a bounded linear operator on a Hilbert space. It is proved that the numerical radius of a pure two-isometry is 1 and the Crawford number of a pure two-isometry is 0. In particular, we show that for any scalar-valuednon-constant inner function $\theta$, the numerical radius and the Crawford number of a Toeplitz operator $T_{\theta}$ on a Hardy space is 1 and 0, respectively. It is also shown that numerical radius is multiplicative for a class of isometries and sub-multiplicative for a class of commutants of a shift. We have illustrated these results with some concrete examples. Finally, some Hardy-type inequalities for Berezin number of certain class of operators are established with the help of the classical Hardy's inequality.