Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator S ∈ B(X) is a generalized inverse of T if TST = T . In this paper we look at several Banach algebras of operators and characterize when an operator in that algebra has a generalized inverse that is also in the algebra. Also, Drazin inverses will be related to generalized inverses and spectral pro...

In this paper, we give explicit expressions of (P ± Q)d of two matrices P and Q, in terms of P, Q, Pd and Qd, under the condition that PQ = P 2, and apply the result to finding an explicit representation for the Drazin inverse of a 2×2 block matrix [ A B C D ] under some conditions.

In this paper, we investigate an explicit representation of the generalized Drazin inverse (a ± b) d in terms of a, a d , b and b d under the condition ab = λba or ab = aba and extend to Banach algebras recent results of C.Y. Deng.

This paper explores links between basis construction methods in Markov decision processes and power series expansions of value functions. This perspective provides a useful framework to analyze properties of existing bases, as well as provides insight into constructing more effective bases. Krylov and Bellman error bases are based on the Neumann series expansion. These bases incur very large in...

The reverse order law and the forward have been studied for various types of generalized inverses. $(b,c)$-inverse is a generalization some well known inverses, such as Moore-Penrose inverse, Drazin core etc. In this paper, $(b,c)$-inverse, in unital ring, investigated an equivalent condition to hold derived. Also, results on topic are generalized. Furthermore, ring with unity introduced, diffe...

Matrix theory and its applications make wide use of the eigenprojections of square matrices. The paper demonstrated that the eigenprojection of a matrix A can be calculated with the use of any annihilating polynomial for A, where u ≥ indA. This enables one to establish the components and the minimum polynomial of A, as well as the Drazin inverse A.

†Department of Mathematics, Southeast University, Nanjing 210096, China ‡College of Mathematics and Information Science, Guangxi University, Nanning 530004, China E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Abstract: It is well known that for an associative ring R, if ab is Drazin invertible then ba is Drazin invertible. In this case, (ba)D = b((ab)D)2a. This for...