Analyticity of the Resolvent Function
Achiles Nyongesa Simiyu *
Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 590-50100, Kakamega, Kenya.
Philis Alosa
Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 590-50100, Kakamega, Kenya.
Fanuel Olege
Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 590-50100, Kakamega, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.
Keywords: Resolvent set, resolvent function, analyticity, spectrum