A note on norm estimates of the numerical radius

Abstract

For a bounded linear operator $A$ on a Hilbert space $\mathcal{H}$, let $\| A \|$ denote the operator norm and $w(A)$ the numerical radius. It is well-known that \begin{equation*} \frac{1}{2} \| A \| ≤q w(A) ≤q \| A \|. \end{equation*} For equalities, we consider linear operators $A$ with $A^{2} = 0$ and normaloid matrices.