Improved bounds for the numerical radius via polar decomposition of operators

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 $\tilde{A}=|A{|}^{1/2}U|A{|}^{1/2}$ is the Aluthge transform of $A$ and $A=U|A|$ is the polar decomposition of $A$. Other related results are also provided.

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