TITLE:
Generalized Invertibility of Operators through Spectral Sets
AUTHORS:
E. Salgado-Matias, S. V. Djordjević, G. Kantún-Montiel
KEYWORDS:
Generalized Inverse, Matrix Form, Resolvent Function, Spectral Projection
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.13 No.2,
February
27,
2024
ABSTRACT: If an operator is not invertible, we are interested if there is a subspace such that the reduction of the operator to that subspace is invertible. In this paper we give a spectral approach to generalized inverses considering the subspace determined by the range of the spectral projection associated with an operator and a spectral set containing the point 0. We compare the cases, 0 is a simple pole of the resolvent function, 0 is a pole of order n of the resolvent function, 0 is an isolated point of the spectrum, and 0 is contained in a circularly isolated spectral set.