On the numerical range of an operator

Abstract

The numerical range of an operator P in a Hubert space is defined as the set of all the complex numbers (Tx, x), where x is a unit vector in the space. It is well known that a bounded normal operator has the property that the closure of its numerical range is exactly the convex hull of its spectrum [5, pp. 325-327, Theorem 8.13 and Theorem 8.14]. Call this property A. In this article let P denote a linear bounded operator in a Hilbert space H, V(T) be its numerical range, K(T) be the convex hull of its spectrum, and use the usual notations p(T), <t(T), Pa(T), Ca(T), Rcr(T) respectively for the resolvent set, spectrum, point spectrum, continuous spectrum,and residual spectrum of P. This article shall investigate some consequences of the property A. In general P having the property A need not be normal. Hence little can be said about an operator unless something else is known. Define a property B if a(T) lies on a convex curve. Note that a class of operators such as unitary operators have this property. The following theorems shall be proved :