Some remarks on invariant subspaces in real Banach spaces (revised version)

TL;DR

This paper establishes conditions under which commutative algebras of operators on real Banach spaces have invariant subspaces, extending to essentially selfadjoint operators on real Hilbert spaces.

Contribution

It proves that certain norm conditions imply the existence of invariant subspaces for commutative operator algebras on reflexive real Banach spaces.

Findings

Invariant subspaces exist under specified norm conditions.

Commutative families of essentially selfadjoint operators have invariant subspaces.

Results apply to operators on real Hilbert spaces.

Abstract

It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators on a real Hilbert space.