Does a typical $\ell_p\,$-$\,$space contraction have a non-trivial invariant subspace?

TL;DR

This paper investigates whether typical contraction operators on classical Banach spaces like nd  have non-trivial invariant subspaces, using Baire category methods in various operator topologies.

Contribution

It provides new results on the prevalence of invariant subspaces among typical contractions in certain topologies, especially the Strong and Strong* Operator Topologies.

Findings

In the Strong Operator Topology, typical contractions have non-trivial invariant subspaces.

In the Strong* Operator Topology, the existence of invariant subspaces for typical contractions is analyzed.

The results depend on the chosen topology and the space nd .

Abstract

Given a Polish topology $\tau$ on ${{\mathcal{B}}_{1}(X)}$, the set of all contraction operators on $X=\ell_p$, $1\le p<\infty$ or $X=c_0$, we prove several results related to the following question: does a typical $T\in {{\mathcal{B}}_{1}(X)}$ in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense $G_\delta$ set $\mathcal G\subseteq ({{\mathcal{B}}_{1}(X)},\tau)$ such that every $T\in\mathcal G$ has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong$^*$ Operator Topology.