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

Grivaux, Sophie; Matheron, Étienne; Menet, Quentin

doi:10.1090/tran/8446

Does a typical $\ell _p\,$-$\,$space contraction have a non-trivial invariant subspace?
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by Sophie Grivaux, Étienne Matheron and Quentin Menet PDF

Trans. Amer. Math. Soc. 374 (2021), 7359-7410 Request permission

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.

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