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arXiv · 2209.10921

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

V. I. Lomonosov·V. S. Shulman
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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.

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V. I. Lomonosov, V. S. Shulman. 2022-09-22. Some remarks on invariant subspaces in real Banach spaces (revised version). https://arxiv.org/abs/2209.10921

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