On mapping theorems for numerical range

الأصل · EN

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 T ≤ 2w(T). We obtain a local improvement of this estimate, namely, if w(T)≤1 then Tx²≤ 2+2√1-| Tx,x|² (x∈ H, x≤1). Using this refinement, we give a simplified proof of Drury's teardrop theorem, which extends the Berger-Stampfli theorem to the case f(0)≠0.