Type: Article
Publication Date: 1972-04-01
Citations: 30
DOI: https://doi.org/10.2140/pjm.1972.41.153
If T is an operator (bounded endormorphism) on the complex Hubert space H, then T e & if and only if 11 (Γziy 1 \ | = l/d(s, W(T)) for all z£Q\ W(T), where Cl W(T) is the closure of the numerical range of T and d(z, W(T)) = inf {\z -u\: ue W(T)}.The main results of this paper are: (1) Te & if and only if the boundary of the numerical range of T is a subset of σ(T), the spectrum of T; and (2) & is an arc-wise connected, closed nowhere dense subset of the set of all operators on H (norm topology) when dim H ^ 2. Introduction* If T is an operator (bounded endomorphism) on the complex Hubert space H, then , σ(T)) ^ || (T -zIΓ\\ and || (T -zIΓ\\ £ l/d(z, W{T)) ,