On the closure of the numerical range of an operator

Abstract

If T is a bounded linear mapping (briefly, operator) in a Hilbert space SC, the numerical range of T is the set W(T) = { (Tx, x): x|| ==1}; thus W(T) is convex [8, p. 131], and its closure cl[W(T)] is compact and convex. Roughly speaking, in this note we observe that cl [W(T)] can be uniquely defined for an element T of an abstract C*-algebra, while W(T) cannot. The C*-algebra setting yields an extension of the spectral convexity theorem [8, p. 327] to nonnormal operators (Corollary 1 of Theorem 3), as well as a reformulation of a theorem of C. Berger (Corollary 2 of Theorem 3).