Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series
Using a Laurent series representation, the authors present a detailed discussion of necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space that are singular in the unperturbed state. In fact, the main purpose of this paper is to complete the analysis of the conditions under which the inverse of a linearly-perturbed operator can be represented by a Laurent series with a finite-order pole at the orgin. The most relevant references are the two papers [\textit{P. Howlett, K. Avrachenkov, C. Pearce} and \textit{V. Ejov}, J. Math. Anal. Appl. 353, No.~1, 68--84 (2009; Zbl 1216.47020); \textit{P. Howlett, A. Albrecht} and \textit{C. Pearce}, J. Math. Anal. Appl. 366, No.~1, 112--123 (2010; Zbl 1192.47010)], where the authors first proposed that the inversion could be expressed in terms of corresponding complementary projections in the domain and range spaces.
scientific article; zbMATH DE number 5925609
scientific article