Invariant subspaces for commuting operators in a real Banach space

TL;DR

This paper proves that certain commutative operator algebras in reflexive real Banach spaces have invariant subspaces under a specific essential norm condition, extending to essentially selfadjoint operators in real Hilbert spaces.

Contribution

It establishes the existence of invariant subspaces for commutative algebras satisfying a new essential norm condition, generalizing previous results.

Findings

Invariant subspaces exist under the given norm condition.

Applies to commutative families of essentially selfadjoint operators.

Extends invariant subspace results to reflexive real Banach spaces.

Abstract

It is proved that a commutative algebra $A$ of operators in 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 in a real Hilbert space.