On Bounded Finite Potent Operators on Arbitrary Hilbert Spaces

Abstract

The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate’s trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.

Drazin-Star and Star-Drazin Inverses of Bounded Finite Potent Operators on Hilbert 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.…”

“…Recently, the author of this work has studied in [14] the set of bounded finite potent endomorphisms on an arbitrary Hilbert space, which will be denoted by B fp (H).…”

“…Accordingly, since U ϕ = u i i≥4 , a non-difficult computation shows that the general solution of (4.2) is [14,Proposition 4.5] one has that W ϕ * = U ⊥ ϕ and from Proposition 2.7 we know that (ϕ * ) † = (ϕ † ) * , one has that…”