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On factorizations of analytic operator-valued functions and eigenvalue multiplicity questions. (English) Zbl 1316.47014
Integral Equations Oper. Theory 82, No. 1, 61-94 (2015); erratum ibid. 85, No. 2, 301-302 (2016).
Summary: 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 Y. Latushkin and A. Sukhtayev [Math. Model. Nat. Phenom. 5, No. 4, 269–292 (2010; Zbl 1193.35110)] by employing a different technique based on factorizations of analytic operator-valued functions due to J. S. Howland [J. Math. Anal. Appl. 36, 12–21 (1971; Zbl 0234.47009)]. 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.

##### MSC:
 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 47A55 Perturbation theory of linear operators
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