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Controlling Almost-Invariant Halfspaces in Both Real and Complex Settings

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Abstract

If T is a bounded linear operator acting on an infinite-dimensional Banach space X, we say that a closed subspace Y of X of both infinite dimension and codimension is an almost-invariant halfspace (AIHS) under T whenever \(TY\subseteq Y+E\) for some finite-dimensional subspace E, or, equivalently, \((T+F)Y\subseteq Y\) for some finite-rank perturbation \(F:X\rightarrow X\). We discuss the existence of AIHS’s for various restrictions on E and F when X is a complex Banach space. We also extend some of these and other results in the literature to the setting where X is a real Banach space instead of a complex one.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, MacEwan University, Edmonton, AB, T5J P2P, Canada

    Adi Tcaciuc

  2. Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL, 60115, USA

    Ben Wallis

Authors
  1. Adi Tcaciuc
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  2. Ben Wallis
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Correspondence to Ben Wallis.

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The authors thank Robert B. Israel for his helpful comments.

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Tcaciuc, A., Wallis, B. Controlling Almost-Invariant Halfspaces in Both Real and Complex Settings. Integr. Equ. Oper. Theory 87, 117–137 (2017). https://doi.org/10.1007/s00020-016-2339-5

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  • DOI: https://doi.org/10.1007/s00020-016-2339-5

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