On numerical ranges of convexoid operators

Numerical range is useful in studying operators on Hilbert spaces. In particular, the geometrical properties of numerical range often provide useful information about algebraic and analytic properties of an operator. The theory of numerical range played a crucial role in the study of some algebraic structures especially in the non-associative context. The numerical range of an operator depends strongly upon the base field. Motivated by theoretical study and applications, researchers have considered different generalizations of numerical range. Numerical range of an operator may be a point, or a line segment containing none, one or all of its end points. Numerical range of another operator may be an open set, closed set or neither. In this paper, we give results of numerical range of convexoid operators. Let be an infinite dimensional complex Hilbert space and be algebra of all bounded linear operators on . is said to be convexoid if the closure of the numerical range coincides with the convex hull of its spectrum. In this paper, we determine the numerical ranges of convexoid operators. We employ some results for convexoid operators due to Furuta and numerical ranges due to Shapiro, Furuta and Nakamoto, Mecheri and Okelo. Some properties of numerical ranges are also discussed.