Numerical range for certain classes of operators

Abstract

We prove a basic result which relates the structure of the spectrum to the interior of the numerical range. Using this result we derive corollaries concerning compact operators, quasi- nilpotents, and finite dimensional operators. In particular, we characterize finite dimensional convexoid operators. 1. Introduction. In this paper will mean a bounded linear transformation of the complex Hilbert space 77 into itself. The thrust of the conclusions that we obtain here is to show that the numerical range of an operator T can be described provided that T—zI has closed range. For a general discussion of numerical range see Chapter 17 of (4). 2. Preliminaries. For an isolated eigenvalue z there are two different notions of eigenspace ; the geometric eigenspace is just the kernel of T—zI (which we simply write as T—z). The algebraic eigenspace associated with z is the range of an idempotent P defined by a contour integral according to the Banach space operational calculus (see pp. 178-181 of (5), for example). If the underlying Hilbert space is H then both F77 and (I—P)H are in- variant under F, and the restriction of T—z to F77 (which we denote T—zjPH) is quasi-nilpotent. We shall say that an eigenvalue is a normal eigenvalue if the corresponding geometric eigenspace reduces F; if a normal eigenvalue is an isolated eigenvalue and the geometric multi- plicity agrees with the algebraic multiplicity then we say that it is a normal-isolated eigenvalue. Clearly an isolated eigenvalue for a normal operator is a normal-isolated eigenvalue. It will be convenient to denote by W(T) the numerical range of F, i.e. {(Tfif): ||/|| = 1}, and the closure of the numerical range is denoted by W(T)~. Finally, an operator is convexoid provided that W(T)~ is the convex hull of the spectrum of T, denoted conv a(T). Note that it follows from Theorem 1.24 on p. 16 of (8) that every point of conv a(T) can be