We study the continuity of the generalized Drazin inverse for elements of Banach algebras and bounded linear operators on Banach spaces. This work extends the results obtained by the second author on the conventional Drazin inverse.
This paper characterizes the Banach algebras of continuous functions on which the spectral factorization mapping 𝔖 is continuous or bounded. It is shown that 𝔖 is continuous if and only if the Riesz projection is bounded on the algebra, and that 𝔖 is bounded only if the algebra is isomorphic to the algebra of continuous functions. Consequently, 𝔖 can never be both continuous and bounded, on any algebra under consideration.
On montre que si est une contraction à spectre dénombrable et telle que, pour tout
The notion of convergence in the generalized sense of a sequence of closed operators is generalized to the situation where the closed operators involved are affiliated with a Banach algebra of operators. Also, the concept of convergence in the generalized sense is extended to the context of a LMC-algebra, where it applies to the spectral theory of the algebra.
The paper concerns an approximation of an eigenvalue problem for two forms on a Hilbert space . We investigate some approximation methods generated by sequences of forms and defined on a dense subspace of . The proof of convergence of the methods is based on the theory of the external approximation of eigenvalue problems. The general results are applied to Aronszajn’s method.