A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces

TL;DR

This paper introduces a comprehensive method for constructing operators lacking non-trivial invariant closed subsets across various Banach spaces, extending Read's previous constructions and exploring their applicability in Hilbertian contexts.

Contribution

It unifies and generalizes existing constructions of invariant subset-free operators on classical Banach spaces and extends these methods to new settings, including quasireflexive dual spaces.

Findings

Constructed operators without non-trivial invariant closed subsets on multiple Banach spaces.

Unified and generalized previous constructions by Read.

Extended methods to the Hilbertian setting and quasireflexive dual spaces.

Abstract

We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces $\ell_{1}$, $c_{0}$ or $\oplus_{\ell_{2}}J$, and without non-trivial invariant subsets on $\ell_{1}$. We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasireflexive dual Banach space which has no non-trivial $w^{*}$-closed invariant subspace.