Open AccessAnalyticity of the Resolvent Function
Author(s) -
Achiles Nyongesa Simiyu,
Philis Alosa,
Olege Fanuel
Publication year - 2021
Publication title -
journal of advances in mathematics and computer science
Language(s) - English
Resource type - Journals
ISSN - 2456-9968
DOI - 10.9734/jamcs/2021/v36i1030407
Subject(s) - resolvent , resolvent formalism , bounded function , mathematics , bounded operator , analytic function , complex plane , banach space , mathematical analysis , operator (biology) , pure mathematics , fredholm theory , spectrum (functional analysis) , function (biology) , fredholm integral equation , integral equation , finite rank operator , physics , biochemistry , chemistry , repressor , quantum mechanics , evolutionary biology , biology , transcription factor , gene
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.