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Brief communications

Invariant Subspaces for Commuting Operators on a Real Banach Space

V. I. Lomonosov^a, V. S. Shul'man<sup>b

a</sup> Department of Mathematics, Kent State University, Kent, USA
^b Department of Higher Mathematics, Vologda State University, Vologda, Russia

Abstract: 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.

UDC: 517

Received: 09.11.2016

DOI: 10.4213/faa3454

 English version: Functional Analysis and Its Applications, 2018, 52:1, 53–56

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