Does a typical ℓ_{𝑝}-space contraction have a non-trivial invariant subspace?

Abstract

Given a Polish topology τ \tau on B 1 ( X ) \mathcal {B}_{1}(X) , the set of all contraction operators on X = ℓ p X=\ell _p , 1 ≤ p > ∞ 1\le p>\infty or X = c 0 X=c_0 , we prove several results related to the following question: does a typical T ∈ B 1 ( X ) 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 δ G_\delta set G ⊆ ( B 1 ( X ) , τ ) \mathcal G\subseteq (\mathcal {B}_{1}(X),\tau ) such that every T ∈ G T\in \mathcal G has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong ∗ ^* Operator Topology.