On mapping theorems for numerical range

Abstract

Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\le 1$. According to a theorem of Berger and Stampfli, if $f$ is a function in the disk algebra such that $f(0)=0$, then $w(f(T))\le \Vert f{\Vert }_{\infty }$. We give a new and elementary proof of this result using finite Blaschke products. A well-known result relating numerical radius and norm says $\Vert T\Vert \le 2w(T)$. We obtain a local improvement of this estimate, namely, if $w(T)\le 1$ then \[ \|Tx\|^2\le 2+2\sqrt{1-|\langle Tx,x\rangle|^2} \qquad(x\in H,~\|x\|\le1). \] Using this refinement, we give a simplified proof of Drury's teardrop theorem, which extends the Berger-Stampfli theorem to the case $f(0)\ne 0$.