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

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.