Local spectral properties of typical contractions on \(\ell_p,\)-$\,$spaces

TL;DR

This paper investigates the local spectral properties of typical contraction operators on p spaces, revealing that most have Dunford's Property (C) but lack other spectral properties, and explores their asymptotic behavior.

Contribution

It provides a Baire category analysis of typical contractions on p spaces, identifying their spectral properties and indecomposability, which is a novel approach in this context.

Findings

Typical contractions have Dunford's Property (C).

Most lack Bishop's Property (B).

Operators are completely indecomposable.

Abstract

We study some local spectral properties of contraction operators on $\ell_p$, $1<p<\infty$ from a Baire category point of view, with respect to the Strong$^*$ Operator Topology. In particular, we show that a typical contraction on $\ell_p$ has Dunford's Property (C) but neither Bishop's Property $(\beta)$ nor the Decomposition Property $(\delta)$, and is completely indecomposable. We also obtain some results regarding the asymptotic behavior of orbits of typical contractions on $\ell_p$.