Refined inequalities for the numerical radius of Hilbert space operators

TL;DR

This paper introduces new, sharper bounds for the numerical radius of operators on Hilbert spaces, improving upon existing inequalities and applying these results to products and commutators of operators.

Contribution

The paper presents novel refined inequalities for the numerical radius, including bounds involving real and imaginary parts, and extends these to products and commutators of operators.

Findings

New lower bounds for the numerical radius involving real and imaginary parts.

Refined inequalities for the numerical radius of operator products.

Improved upper bounds for the numerical radius of operator commutators.

Abstract

We present some new upper and lower bounds for the numerical radius of bounded linear operators on a complex Hilbert space and show that these are stronger than the existing ones. In particular, we prove that if $A$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$ and if $\Re(A)$, $\Im(A)$ are the real part, the imaginary part of $A$, respectively, then $$ w(A)\geq\frac{\|A\|}{2} +\frac{1}{2\sqrt{2}} \Big | \|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\| \Big | $$ and $$ w^2(A)\geq\frac{1}{4}\|A^*A+AA^*\|+\frac{1}{4}\Big| \|\Re(A)+\Im(A)\|^2-\|\Re(A)-\Im(A)\|^2\Big|. $$ Here $w(.)$ and $\|.\|$ denote the numerical radius and the operator norm, respectively. Further, we obtain refinement of inequalities for the numerical radius of the product of two operators. Finally, as an application of the second inequality mentioned above, we obtain an improvement of upper bound for the numerical radius of the commutators of operators.