Generic properties of l_p-contractions and similar operator topologies

TL;DR

This paper investigates how the notion of generic properties of contraction operators on certain Banach spaces depends on the choice of topology, showing that for specific spaces and topologies, the same comeager sets are shared.

Contribution

It demonstrates that for $ ext{l}_p$ spaces with $p=3$ or $3/2$, various natural topologies on contraction operators have identical generic properties, revealing topology-independent genericity.

Findings

For $ ext{l}_p$ with $p=3$ or $3/2$, certain topologies share the same comeager sets.

The study identifies conditions under which generic properties are topology-independent.

Analysis of continuity points and norming vectors is key to understanding these properties.

Abstract

If $X$ is a separable reflexive Banach space, there are several natural Polish topologies on $\mathcal{B}(X)$, the set of contraction operators on $X$ (none of which being clearly ``more natural'' than the others), and hence several a priori different notions of genericity -- in the Baire category sense -- for properties of contraction operators. So it makes sense to investigate to which extent the generic properties, i.e. the comeager sets, really depend on the chosen topology on $\mathcal{B}(X)$. In this paper, we focus on $\ell_p\,$-$\,$spaces, $1<p\neq 2<\infty$. We show that for some pairs of natural Polish topologies on $\mathcal B_1(\ell_p)$, the comeager sets are in fact the same; and our main result asserts that for $p=3$ or $3/2$ and in the real case, all topologies on $\mathcal B_1(\ell_p)$ lying between the Weak Operator Topology and the Strong$^*$ Operator Topology share the same comeager sets. Our study relies on the consideration of continuity points of the identity map for two different topologies on $\mathcal{B}_1 (\ell_p)$. The other essential ingredient in the proof of our main result is a careful examination of norming vectors for finite-dimensional contractions of a special type.