A reflexive HI space with the hereditary Invariant Subspace Property

TL;DR

This paper constructs a reflexive hereditarily indecomposable Banach space where every bounded linear operator on any infinite dimensional subspace has a non-trivial closed invariant subspace, advancing the understanding of invariant subspace properties.

Contribution

It introduces a new reflexive hereditarily indecomposable Banach space with the hereditary invariant subspace property for all bounded operators.

Findings

Every bounded operator on infinite dimensional subspaces has a non-trivial invariant subspace.

The space is reflexive and hereditarily indecomposable.

The construction demonstrates the hereditary invariant subspace property in a new class of Banach spaces.

Abstract

A reflexive hereditarily indecomposable Banach space $\mathfrak{X}_{_{^\text{ISP}}}$ is presented, such that for every $Y$ infinite dimensional closed subspace of $\mathfrak{X}_{_{^\text{ISP}}}$ and every bounded linear operator $T:Y\rightarrow Y$, the operator $T$ admits a non-trivial closed invariant subspace.