Comments on the cosmic convergence of nonexpansive maps

This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $$\ell ^{1}$$ ℓ 1 . We also point out some inaccurate assertions appearing in the literature on this topic.

On Bounded Finite Potent Operators on Arbitrary Hilbert Spaces

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.

Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

Let T:H→H be a bounded linear operator on a separable Hilbert space H. In this paper, we construct an isomorphism Fxx*:L2(σ(|T−a|),μ|T−a|,ξ)→L2(σ(|(T−a)*|),μ|(T−a)*|,Fxx*Hξ) such that (Fxx*)2=identity and Fxx*H is a unitary operator on H associated with Fxx*. With this construction, we obtain a noncommutative functional calculus for the operator T and Fxx*=identity is the special case for normal operators, such that S=R|(S−a)|,ξ(Mzϕ(z)+a)R|S−a|,ξ−1 is the noncommutative functional calculus of a normal operator S, where a∈ρ(T), R|T−a|,ξ:L2(σ(|T−a|),μ|T−a|,ξ)→H is an isomorphism and Mzϕ(z)+a is a multiplication operator on L2(σ(|S−a|),μ|S−a|,ξ). Moreover, by Fxx* we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class BLeb(H)⊂B(H) such that T is Li-Yorke chaotic if and only if T*−1 is for a Lebesgue operator T.