Functional Analysis and Its Applications

, Volume 52, Issue 1, pp 53–56 | Cite as

Invariant Subspaces for Commuting Operators on a Real Banach Space

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Abstract

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator TA satisfies the condition
$${\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,$$
where ║ · ║ 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.

Key words

Banach space algebra of operators invariant subspace 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKent State UniversityKentUSA
  2. 2.Department of Higher MathematicsVologda State UniversityVologdaRussia

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