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Generalized Invertibility of Operators through Spectral Sets

DOI: 10.4236/alamt.2023.132002, PP. 21-35

Keywords: Generalized Inverse, Matrix Form, Resolvent Function, Spectral Projection

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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.

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