Javad Mashreghi, Université Laval, Canada

On mapping theorems for numerical range

Abstract

Let T be an operator on a Hilbert space H with numerical radius w(T) ≤ 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)) ≤ |f|_∞. We give a new and elementary proof of this result using finite Blaschke products.

A well-known result relating numerical radius and norm says ≤ 2w(T). We obtain a local  improvement of this estimate, namely, if w(T) ≤ 1 then

Using this refinement, we give a simplified proof of Drury’s teardrop theorem, which extends the Berger-Stampfli theorem to the case f(0) ¹ 0.

Joint work with T. Ransford and H. Klaja.