Numerical radius of certain two-by-two block matrices

Abstract

We investigate the numerical range $W(T)$ and numerical radius $w(T)$ of operators of the form $T=\left(\begin{array}{cc}A& B\\ 0& 0\end{array}\right)$. We show that $W(T)$ is the union of the numerical ranges of a family of $2\times 2$ matrices, $T_x$, leading to several consequences, including improved inequalities for $w(T)$. For cases where $A$ is a self-adjoint involution, we characterize the conditions under which $W(T)$ is an elliptical disk and determine the minimum numerical radius of $T_U=\left(\begin{array}{cc}{U}^{\ast }AU& B\\ 0& 0\end{array}\right)$ over all unitary operators $U$. Finally, we study matrices $T\in M_n$ satisfying $\Vert T^{m}x\Vert =\Vert T^m\Vert =\Vert T\Vert$ for a unit vector $x$ and all positive integers $m$. This analysis connects these matrices to the aforementioned block form and provides a counterexample to the conjecture that if $\Vert T^k\Vert =\Vert T\Vert$ for all $k\ge 1$, then some power of the matrix has a direct summand that is a scalar multiple of an idempotent.

28 pages