We consider a ψ-irreducible, discrete-time Markov chain on a general state space with transition kernel P . Under suitable conditions on the chain, kernels can be treated as bounded linear operators between spaces of functions or measures and the Drazin inverse of the kernel operator I−P exists. The Drazin inverse provides a unifying framework for objects governing the chain. This framework is ...

is called the Drazin inverse of A and is denoted by X = A, where k is the index of A, i.e., the smallest non-negative integer such that rank(A) = rank(A). We denote such a k by Ind(A). It is well-known that A exists and is unique (see [2]). If Ind(A) = 1, A is also called the group inverse of A and is denoted by A. Then A exists if and only if rank(A) = rank(A) (see [1, 3, 11-14, 24, 25, 29]). ...

The purpose of this paper is to introduce a new generalized inverse, called DMP inverse, associated with a square complex matrix using its Drazin and Moore-Penrose inverses. DMP inverse extends the notion of core inverse, introduced by O.M. Baksalary and G. Trenkler for matrices of index at most 1 in [Core inverse of matrices, Linear and Multilinear Algebra, 2010, 681–697] to matrices of an arb...

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [15], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more effective estimation of degrees of polynomial matrices generated in Leverrier-Faddev method. Algorithms are implemented and tested i...

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...