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​​​​​International Journal of Modern Science and Technology, Vol. 2, No. 2, 2017, Pages 56-60.

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​On Numerical Ranges of Convexoid Operators

J. A. Otieno, N. B. Okelo*, O. Ongati
School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology,
P.O Box 210-40601, Bondo-Kenya.
*Corresponding author’s e-mail: bnyaare@yahoo.com

​Abstract
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.

​​Keywords: Numerical range; Convexoid operator; Numerical radius; Spectrum.

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ISSN 2456-0235

International Journal of Modern Science and Technology