In this article the (b, c)-inverse will be studied. Several equivalent conditions for the existence of the (b, c)-inverse in rings will be given. In particular, the conditions ensuring the existence of the (b, c)-inverse, of the annihilator (b, c)-inverse and of the hybrid (b, c)-inverse will be proved to be equivalent, provided b and c are regular elements in a unitary ring R. In addition, the set of all (b, c)-invertible elements will be characterized and the reverse order law will be also studied. Moreover, the relationship between the (b, c)-inverse and the Bott-Duffin inverse will be considered. In the context of Banach algebras, integral, series and limit representations will be given. Finally the continuity of the (b, c)-inverse will be characterized.Mathematics Subject Classification. 15A09, 16B99, 16U99, 46H05.
Abstract. Elements with equal idempotents related to their image-kernel (p, q)-inverses are characterized, and applications to perturbations, reverse order law and commutativity are given.
In this note we consider the problem of localization and approximation of
eigenvalues of operators on infinite dimensional Banach and Hilbert spaces.
This problem has been studied for operators of finite rank but it is seldom
investigated in the infinite dimensional case. The eigenvalues of an operator
(between infinite dimensional vector spaces) can be positioned in different
parts of the spectrum of the operator, even it is not necessary to be
isolated points in the spectrum. Also, an isolated point in the spectrum is
not necessary an eigenvalue. One method that we can apply is using Weyl?s
theorem for an operator, which asserts that every point outside the Weyl
spectrum is an isolated eigenvalue.
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