Development of inequality and characterization of equality conditions for the numerical radius

Abstract

Let $A$ be a bounded linear operator on a complex Hilbert space and $\Re (A)$ ( $\Im (A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that \begin{eqnarray*} w(A)&\geq &\frac{1}{2} \left \|A \right\| + \frac{ 1}{2} \mid \|\Re(A)\|-\|\Im(A)\|\mid,\,\,\mbox{and}\\ w^2(A)&\geq& \frac{1}{4} \left \|A^*A+AA^* \right\| + \frac{1}{2}\mid \|\Re(A)\|^2-\|\Im(A)\|^2 \mid, \end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=\frac{1}{2}\sqrt{\Vert A^{\ast }A+A{A}^{\ast }\Vert }$ and prove that $w(A)=\frac{1}{2}\sqrt{\Vert A^{\ast }A+A{A}^{\ast }\Vert }$ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $\frac{1}{2}\sqrt{\Vert A^{\ast }A+A{A}^{\ast }\Vert }$. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.

10 pages

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