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

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 $$\mathrm{\parallel }1-\epsilon T^2{\mathrm{\parallel }}_e\le 1+o(\epsilon )\text{ when }\epsilon \searrow 0,$$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_e$ 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.