Perturbacije uopštenih inverza elemenata u prstenima

Abstract

The topic of this doctoral dissertation is the study of some perturbation properties of generalized inverses of the elements in rings with or without involution, as well as in Banach algebra and C*-algebra. So far, the problems are mostly investigated in the case of the matrices, i.e. operators on finite dimensional vector spaces. The idea was to use decompositions of elements in rings induced by idempotents, instead of matrix decompositions. In this way, the calculus in rings approaches to the calculus in the set of 2x2 matrices over mentioned ring. The additional difficulty appears in using idempotents in Banach and C*-algebras. Starting from the matrix form of an arbitrary ring element regarding to a corresponding idempotents, the matrix forms of the group and core inverse with respect to the same idempotents are constructed. Several characterizations are established for the forward order law for the Moore-Penrose inverse to hold. The results valid for matrices have been extended to elements of arbitrary Banach algebras. Further, we introduced the acute perturbation for the group inverse in Banach algebra with the respect to the spectral radius instead of the spectral the norm. In this way, we described an explicit expression that connects the group inverses of the initial and the perturbed elements. Moreover, more simple expressions for the perturbation of the generalized inverses are obtained and the optimal perturbation bounds for the core inverse presented when the perturbation satisfies certain conditions.