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On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

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An Erratum to this article was published on 22 April 2016

Abstract

We study several natural multiplicity questions that arise in the context of the Birman–Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic operator-valued functions due to Howland. Factorizations of analytic operator-valued functions are of particular interest in themselves and again we re-derive Howland’s results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic operator-valued functions, we recall the notion of the index of a finitely meromorphic operator-valued function and use that to prove an analog of the well-known Weinstein–Aronszajn formula relating algebraic multiplicities of the underlying unperturbed and perturbed operators. Finally, we consider pairs of projections for which the difference belongs to the trace class and relate their Fredholm index to the index of the naturally underlying Birman–Schwinger operator.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Fritz Gesztesy

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Helge Holden

  3. Mathematics Department, The University of Tennessee at Chattanooga, 415 EMCS Building, Dept. 6956, 615 McCallie Ave, Chattanooga, TN, 37403, USA

    Roger Nichols

Authors
  1. Fritz Gesztesy
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  2. Helge Holden
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  3. Roger Nichols
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Correspondence to Fritz Gesztesy.

Additional information

Supported in part by the Research Council of Norway.

R. Nichols gratefully acknowledges support from an AMS–Simons Travel Grant.

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Gesztesy, F., Holden, H. & Nichols, R. On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions. Integr. Equ. Oper. Theory 82, 61–94 (2015). https://doi.org/10.1007/s00020-014-2200-7

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