On the Drazin inverse of finite potent endomorphisms

Drazin-Star and Star-Drazin Inverses of Bounded Finite Potent Operators on Hilbert Spaces

“…One has that i(ϕ) = 0 if and only if V is a finite-dimensional vector space and ϕ is an automorphism. In [16,Lemma 3.2] is proved that for finite-dimensional vector spaces, this definition of index coincides with Definition 2.1 for matrices associated with endomorphisms of finite-dimensional vector spaces.…”

“…Thus, Yekutieli in [21] and Braunling in [2] and [3] have addressed problems of arithmetic symbols by using properties of finite potent endomorphism; Debry in [7] and Taelman in [19] have offered results about Drinfeld modules from these linear operators; and Cabezas Sánchez and Pablos Romo have given explicit solutions of infinite linear systems from reflexive generalized inverses of finite potent endomorphisms in [4]. Moreover, the author of this work has extended to finite potent endomorphisms the notions of Drazin inverse, Group inverse and DMP inverses in [13], [15] and [16] and, recently, has studied the properties of bounded finite potent operators on Hilbert spaces in [14]. As far as we know, this last paper is the first approach for studying finite potent endomorphisms from the point of view of the Functional Analysis that has appeared in the literature.…”

On Bounded Finite Potent Operators on Arbitrary Hilbert Spaces

“…According to [11,Theorem 3.2], if ϕ D is the Drazin inverse of ϕ offered in [13], one has that ϕ 1 = ϕ • ϕ D • ϕ is the core part of ϕ. Also, ϕ 2 is named the nilpotent part of ϕ and one has that…”

“…(Final Remark). During the past few years the author of this work has extended several generalized inverses of finite square complex matrices to finite potent endomorphisms on infinite-dimensional inner product spaces in [11][12][13]. From the results of Sect.…”

On Bounded Finite Potent Operators on arbitrary Hilbert Spaces

“…Let V be again an arbitrary k-vector space. Given a finite potent endomorphism ϕ ∈ End k (V ), there exists a unique decomposition ϕ = ϕ 1 + ϕ 2 , where ϕ 1 , ϕ 2 ∈ End k (V ) are finite potent endomorphisms satisfying that: [11,Theorem 3.2], if ϕ D is the Drazin inverse of ϕ offered in [13], one has that ϕ 1 = ϕ • ϕ D • ϕ is the core part of ϕ. Also, ϕ 2 is named the nilpotent part of ϕ and one has that…”