A reflexive hereditarily indecomposable space with the hereditary invariant subspace property

Abstract

A separable Banach space X satisfies the invariant subspace property (ISP) if every bounded linear operator T ∈ L(X) admits a non-trivial closed invariant subspace. In this paper, we present the first known example of a reflexive Banach space X ISP satisfying the ISP. Moreover, this is the first known example of a Banach space satisfying the hereditary ISP, namely every infinitedimensional subspace of it satisfies the ISP. The space X ISP is hereditarily indecomposable (HI) and every operator T ∈ L(X ISP) is of the form λI + S with S a strictly singular operator. The critical property of the strictly singular operators of X ISP is that the composition of any three of them is a compact one. The construction of X ISP is based on saturation methods and it uses as an unconditional frame Tsirelson space. The new ingredient in the definition of the space is the saturation under constraints, a method initialized in a fundamental work of Edward Odell and Thomas Schlumprecht.

The scalar-plus-compact property in spaces without reflexive subspaces

“…It is based on (19) and Lemma 4.14. Its proof is very similar to that of [AM1,Lemma 3.7], however we include it for completeness.…”

“…The following Lemma is proved using Lemmas 4.15 and 4.16 and arguments very similar to those used in the proof of [AM1,Lemma 3.8]. We include a proof for completeness.…”

“…A tool that has been used in recent constructions involving saturation under constraints is the α-index of a block sequence ( [ABM], [AM1], [AM2] and more). This index helps characterize what spreading models a given block sequence admits.…”

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

“…We are especially interested in determining which geometric properties of a Banach space will ensure that it supports an operator without non-trivial invariant closed subset. This is of interest in the view of the recent works of Argyros and Haydon [1] and Argyros and Motakis [2]: in [1], examples are constructed of spaces on which every operator is of the form λI + K, with λ a scalar and K a compact operator, and it is well known that such operators always have a non-trivial closed invariant subspace [17]. In [2], the authors exhibit reflexive Banach spaces on which every operator has a non-trivial invariant closed subspace.…”

“…This is of interest in the view of the recent works of Argyros and Haydon [1] and Argyros and Motakis [2]: in [1], examples are constructed of spaces on which every operator is of the form λI + K, with λ a scalar and K a compact operator, and it is well known that such operators always have a non-trivial closed invariant subspace [17]. In [2], the authors exhibit reflexive Banach spaces on which every operator has a non-trivial invariant closed subspace. The spaces of [1,2] are hereditarily indecomposable, so they are certainly very far from the kind of spaces which we are going to consider in our forthcoming Theorem 1.1.…”