Core-Nilpotent decomposition and new generalized inverses of finite potent endomorphisms

On Bounded Finite Potent Operators on Arbitrary Hilbert Spaces

“…From the uniqueness of the CN-decomposition ϕ = ϕ 1 +ϕ 2 proved in [11,Theorem 3.2], if ϕ = ψ + φ as in Theorem 3.7, one has that ψ = ϕ 1 and φ = ϕ 2 .…”

“…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.…”

“…In Sect. 2 we recall the basic definitions of this work (inner product spaces, Hilbert spaces, bounded operators, orthogonality and the adjoint of a bounded linear map) and a summary of statements of the articles [1,11,19].…”

“…Finally, Sect. 4 is devoted to offer the characterization of the adjoint ϕ * of a bounded finite potent ϕ ∈ End C (H) from the AST-decomposition of H introduced in [1], the CN-decomposition of ϕ * given in [11], the structure of the spectrum of ϕ * and the relation between the trace of ϕ * and the determinant of Id + ϕ * with the trace of ϕ and the determinant of Id + ϕ respectively.…”