Invariant Subspaces for Commuting Operators on a Real Banach Space

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

$${left| {1 - varepsilon {T^2}} right|_e} leqslant 1 + oleft( varepsilon right)asvarepsilon searrow 0,$$

where ║ · ║ _e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.