Equality of three numerical radius inequalities
Let $A$ be an $n\times n$ matrix over the field of complex numbers and let $w(A)$ and $\| A\| $ numerical radius and operator norm respectively. The following inequalities are well-known: \[ w(A)\le\| A\|, \] \[ w(A)^2\le\frac{\| A\| ^2 +w(A^2)}{2}, \] \[ w(A)\le\frac{ \| A\| +\| A^2\| ^{1/2}}{2}, \] \[ w(A)^2\le\frac{ \| A\| + w(\Delta _t(A))}{2}, \] with $\Delta_t(A)=|A|^tV|A|^{1-t}$, $0\le t\le 1$, being the generalized Aluthge transform of $A$ where $|A|=(A^*A)^{1/2}$ and $V$ is the partial isometry defined on $\mathcal{C}^n=\operatorname{ran }A^*\oplus\operatorname{ker }A$ by $V(|A|x)=Ax$ for any vector $x$ in $\mathcal{C}^n$. In this paper the authors, by imposing certain conditions on matrix $A$, prove equalities for the last three inequalities. More specifically, they prove that \[ w(A)^2=\frac{ \| A\| ^2 +w(A^2)}{2}, \] \[ w(A)=\frac{ \| A\| +\| A^2\| ^{1/2}}{2}, \] \[ w(A)^2=\frac{ \| A\| + w(\Delta _t(A))}{2}. \] They also remark that these three equalities are true for bounded linear operators on a Hilbert space.
scientific article; zbMATH DE number 6912318
scientific article