Invariant subspace problem for reflexive Banach and Hilbert spaces

Determine whether every bounded linear operator on an infinite-dimensional separable reflexive complex Banach space, in particular on a separable Hilbert space, has a non-trivial closed invariant subspace.

Background

The classical invariant subspace problem asks whether every bounded linear operator on a Banach space admits a non-trivial invariant subspace. While counterexamples are known on some non-reflexive spaces (e.g., Enflo and Read), the situation for reflexive spaces—and notably for Hilbert spaces—has resisted resolution.

This paper studies typical properties of positive contractions but recalls the status of the general invariant subspace problem to contextualize the results.

References

The invariant subspace problem still remains open for reflexive Banach spaces and in particular for separable Hilbert spaces.

— Typical properties of positive contractions and the invariant subspace problem  (2409.14481 - Gillet, 2024) in Introduction

It is worthwhile to mention the intimate relationship between this generation question and the well-known and still unsolved Invariant Subspace Problem: does every bounded linear operator on a (separable, complex, infinite-dimensional) Hilbert space H have an invariant (closed) subspace other than the trivial ones {0} and H?

— Fully non-zero matrices and generators of full algebras of operators  (2608.21154 - Marcoux et al., 21 Aug 2026) in Section 1, Introduction, p. 2

For a bounded operator $T$ on a Hilbert space $H$, the invariant subspace problem asks whether there always exists a non-trivial closed subspace $S\subseteq H$ such that $TS\subseteq S$. For infinite-dimensional separable Hilbert spaces, this problem is still open.

— Noncommutative commutant lifting, interpolation, and shift-invariant subspaces  (2609.11410 - Barik et al., 10 Sep 2026) in Section 1, Introduction

The Invariant Subspace Problem is one of the best-known unsolved problems in functional analysis: does every bounded operator on $$ have a nontrivial closed invariant subspace? In the 1930s, von Neumann found a beautiful technique for producing invariant subspaces of compact operators, which Aronszajn and Smith extended to Banach spaces and Lomonosov developed into the statement that every operator commuting with a nonzero compact operator has a nontrivial hyperinvariant subspace. The answer is negative for general Banach spaces, by the examples of Enflo and Read ; for operators on $$ the problem remains open, although positive results are known for many classes, such as hyponormal operators with thick spectra .

— The invariant subspace problem and Rosenblum operators II  (2610.11202 - Fang et al., 8 Oct 2026) in Section 1, introductory discussion of the Invariant Subspace Problem

Whether every tridiagonal operator has a nontrivial closed invariant subspace nevertheless remains open; the theorem below provides another affirmative case.

— The invariant subspace problem and Rosenblum operators II  (2610.11202 - Fang et al., 8 Oct 2026) in Section 7, introductory paragraph of Section \ref{S:tridiagonal}