Numerical radius inequalities for tensor product of operators

TL;DR

This paper introduces new bounds for the numerical radius of tensor products of operators on Hilbert spaces, improving understanding of their spectral properties and conditions for equality.

Contribution

The paper develops novel lower and upper bounds for the numerical radius of tensor product operators and analyzes the conditions under which these bounds are tight.

Findings

New bounds for $w(A ensor B)$ established.

Equality conditions for the bounds characterized.

Enhanced spectral analysis of tensor product operators.

Abstract

The two well-known numerical radius inequalities for the tensor product $A \otimes B$ acting on $\mathbb{H} \otimes \mathbb{K}$, where $A$ and $B$ are bounded linear operators defined on complex Hilbert spaces $\mathbb{H} $ and $ \mathbb{K},$ respectively are, $ \frac{1}{2} \|A\|\|B\| \leq w(A \otimes B) \leq \|A\|\|B\| $ and $w(A)w(B) \leq w(A \otimes B) \leq \min \{ w(A) \|B\|, w(B) \|A\| \}. $ In this article we develop new lower and upper bounds for the numerical radius $w(A \otimes B)$ of the tensor product $A \otimes B $ and study the equality conditions for those bounds.