Improved bounds for the numerical radius via polar decomposition of operators (2303.03051v1)

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.