On The Geometric Properties Of The Numerical Ranges Of Bounded Linear Operators

Abstract

This study is on the characterization of the numerical ranges of a bounded linear operator T on a Hilbert space H. In case of a bounded linear operator, the closure of the numerical range apart from including the spectrum of the operator turns out to be a convex subset of the complex plane. We provide essential expository material on numerical ranges and then proceed towards investigation of some signi cant aspects described below which are the highlights of the study. First, we give a set of su¢ cient conditions for the numerical range of an operator to be closed. For bounded linear operator T which is hyponormal we show that Conv (T) W(T) and (T) W(T): Secondly, we provide a proof to show that the numerical range of an operator on a 2-dimensional complex Hilbert space is, in general, an ellipse. Thirdly, We also consider some special points on the boundary of the numerical range and in this connection shows that: If the numerical range is closed and 2 W(T) such that the boundary of the numerical range is not a di⁄erentiable arc at , then belongs to the point spectrum; if is just a corner of the numerical range, then belongs to the spectrum. If 2 W(T) is such that 2 D for a circular disc D of C and Dn W(T) = f g, then 2 P

(T) In our study and pursuit of these investigations, we expose many minor results and observations which are of interest in themselves.