Improved bounds for the numerical radius via polar decomposition of operators

TL;DR

This paper derives new and improved bounds for the numerical radius of bounded linear operators on complex Hilbert spaces using polar decomposition, spectral radius, and Aluthge transform techniques.

Contribution

It introduces novel inequalities for the numerical radius that generalize and improve previous bounds by leveraging polar decomposition and related operator transforms.

Findings

Established new upper bounds involving the spectral radius and Aluthge transform.

Generalized existing inequalities for the numerical radius with tighter bounds.

Provided several related inequalities and bounds for operator norms.

Abstract

Using the polar decomposition of a bounded linear operator $A$ defined on a complex Hilbert space, we obtain several numerical radius inequalities of the operator $A$, which generalize and improve the earlier related ones. Among other bounds, we show that if $w(A)$ is the numerical radius of $A$, then \begin{eqnarray*} w(A) &\leq& \frac12 \|A\|^{1/2} \left\| |A|^{t} + |A^*|^{1-t} \right \|, \end{eqnarray*} for all $t\in [0,1].$ Also, we obtain some upper bounds for the numerical radius involving the spectral radius and the Aluthge transform of operators. It is shown that \begin{eqnarray*} w(A) &\leq& \|A\|^{1/2} \left( \frac12 \left \| \frac{ |A|+|A^*|}2 \right\| +\frac12 \left\| \widetilde{A}\right \| \right)^{1/2}, \end{eqnarray*} where $\widetilde{A}= |A|^{1/2}U|A|^{1/2} $ is the Aluthge transform of $A$ and $A=U|A|$ is the polar decomposition of $A$. Other related results are also provided.