It is proved that a commutative algebra A of operators on a reflexive real
Banach space has an invariant subspace if each operator T∈A satisfies the
condition ∥1−εT2∥e​≤1+o(ε) when ε↘0, where ∥⋅∥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