Improved numerical radius bounds using the Moore-Penrose inverse

Abstract

Using the Moore-Penrose inverse of a bounded linear operator, we obtain few bounds for the numerical radius, which improve the classical ones. Applying these improvements, we study equality conditions of the existing bounds. It is shown that if T is a bounded linear operator with closed range, then$w^2\left(T\right)\le \frac{1}{2}\Vert T^{⁎}T+T{T}^{⁎}\Vert (\frac{1}{2}\Vert T{T}^{†}+T^{†}T\Vert )\le \frac{1}{2}\Vert T^{⁎}T+T{T}^{⁎}\Vert .$ For a finite-dimensional space operator T, this improvement is proper if and only if $Range\left(T\right)\cap Range\left(T^{⁎}\right)=\left\{0\right\}$. Clearly, if $\Vert T{T}^{†}+T^{†}T\Vert =1$, then $w^2\left(T\right)=\frac{1}{4}\Vert T^{⁎}T+T{T}^{⁎}\Vert$. Among other results, we obtain inner product inequalities for the sum of operators, and as an application of these inequalities, we deduce relevant operator norm and numerical radius bounds.