English

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

Functional Analysis 2022-09-23 v1

Abstract

It is proved that a commutative algebra AA of operators on a reflexive real Banach space has an invariant subspace if each operator TAT\in A satisfies the condition 1εT2e1+o(ε) when ε0,\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0, where e\|\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.

Keywords

Cite

@article{arxiv.2209.10921,
  title  = {Some remarks on invariant subspaces in real Banach spaces (revised version)},
  author = {V. I. Lomonosov and V. S. Shulman},
  journal= {arXiv preprint arXiv:2209.10921},
  year   = {2022}
}
R2 v1 2026-06-28T01:53:19.522Z