Improvement of numerical radius inequalities

TL;DR

This paper introduces improved bounds for the numerical radius of 2x2 off-diagonal operator matrices and extends inequalities for bounded linear operators on complex Hilbert spaces, enhancing existing results.

Contribution

It generalizes and refines numerical radius inequalities, providing tighter bounds and new inequalities for operators on Hilbert spaces.

Findings

Derived new upper and lower bounds for numerical radius of operator matrices.

Established a generalized inequality for the numerical radius involving operator powers.

Improved existing numerical radius inequalities for specific cases like r=1.

Abstract

We develop upper and lower bounds for the numerical radius of $2\times 2$ off-diagonal operator matrices, which generalize and improve on the existing ones. We also show that if $A$ is a bounded linear operator on a complex Hilbert space and $|A|$ stands for the positive square root of $A$, i.e., $|A|=(A^*A)^{1/2}$, then for all $r\geq 1$, $w^{2r}(A) \leq \frac{1}{4} \big \| |A|^{2r}+|A^*|^{2r} \big \| + \frac{1}{2} \min\left\{ \big \|\Re\big(|A|^r\, |A^*|^r \big) \big \|, w^r(A^2) \right\} $ where $w(A)$, $\|A\|$ and $\Re(A)$, respectively, stand for the numerical radius, the operator norm and the real part of $A$. This (for $r=1$) improves on existing well-known numerical radius inequalities.