Does a typical ${\ell }_p$-space contraction have a non-trivial invariant subspace?

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${}^{\ast }$ Operator Topology.

50 pages

References in corpus (2)

• Linear dynamical systems on Hilbert spaces: typical properties and explicit examples
• Constrained von Neumann inequalities