On mapping theorems for numerical range

TL;DR

This paper provides a new proof of a theorem relating the numerical radius of an operator to functions in the disk algebra, introduces a local improvement of the norm estimate based on the numerical radius, and simplifies the proof of Drury's teardrop theorem extending the original result.

Contribution

It offers an elementary proof of Berger-Stampfli's theorem using Blaschke products, introduces a refined local estimate for the norm of operators with bounded numerical radius, and simplifies the proof of Drury's teardrop theorem.

Findings

New elementary proof of Berger-Stampfli theorem using finite Blaschke products.

A local improvement of the norm estimate for operators with numerical radius ≤ 1.

Simplified proof of Drury's teardrop theorem extending the original result.

Abstract

Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\le1$. According to a theorem of Berger and Stampfli, if $f$ is a function in the disk algebra such that $f(0)=0$, then $w(f(T))\le\|f\|_\infty$. We give a new and elementary proof of this result using finite Blaschke products. A well-known result relating numerical radius and norm says $\|T\| \leq 2w(T)$. We obtain a local improvement of this estimate, namely, if $w(T)\le1$ 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 $f(0)\ne0$.