Geodesic
Invariant Subspaces for Commuting Operators on a Real Banach Space
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 1, pp. 65-69

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It is proved that the commutative algebra $\mathcal{A}$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in\mathcal{A}$ satisfies the condition $$ \|1-\varepsilon T^2\|_e\le 1+o(\varepsilon)\ \text{as}\ \varepsilon\searrow 0 $$ where $\|\cdot\|_e$ 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.
Keywords: Banach space, algebra of operators, invariant subspace. 
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     author = {V. I. Lomonosov and V. S. Shul'man},
     title = {Invariant {Subspaces} for {Commuting} {Operators} on a {Real} {Banach} {Space}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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V. I. Lomonosov; V. S. Shul'man. Invariant Subspaces for Commuting Operators on a Real Banach Space. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 1, pp. 65-69. http://geodesic.mathdoc.fr/item/FAA_2018_52_1_a6/