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The Selgrade Decomposition for Linear Semiflows on Banach Spaces

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Abstract

We extend Selgrade’s Theorem, Morse spectrum, and related concepts to the setting of linear skew product semiflows on a separable Banach bundle. We recover a characterization, well-known in the finite-dimensional setting, of exponentially separated subbundles as attractor–repeller pairs for the associated semiflow on the projective bundle.

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Notes

  1. What is meant by this shorthand is that if \(N = \infty \), then \(i \in \mathbb {N}\), and if \(N < \infty \), then \(1 \le i \le N\)

  2. See Lemmas 3.8 and 4.9. We note however that continuity can be deduced directly from exponential separation, as carried out in, e.g., [4]

  3. We use the terminology ‘discrete’ to evoke an analogy with the discrete spectrum of a closed linear operator.

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Authors and Affiliations

  1. University of Maryland, College Park, 4305 Kirwan Hall, College Park, MD, 20742, USA

    Alex Blumenthal

  2. University of Missouri, 104 Mathematics Bldg., Columbia, MO, 65211, USA

    Yuri Latushkin

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  1. Alex Blumenthal
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  2. Yuri Latushkin
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Correspondence to Yuri Latushkin.

Additional information

Dedicated to the memory of George Sell, to whom we owe so much.

Alex Blumenthal: This material is based upon work supported by the National Science Foundation under Award No. 1604805. Yuri Latushkin: Partially supported by the NSF Grants DMS-1067929 and DMS-1710989, by the Research Board and Research Council of the University of Missouri, and by the Simons Foundation.

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Blumenthal, A., Latushkin, Y. The Selgrade Decomposition for Linear Semiflows on Banach Spaces. J Dyn Diff Equat 31, 1427–1456 (2019). https://doi.org/10.1007/s10884-018-9648-0

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  • DOI: https://doi.org/10.1007/s10884-018-9648-0

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