Integral Equations and Operator Theory

, Volume 82, Issue 1, pp 61–94 | Cite as

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

Article

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.

Mathematics Subject Classification

Primary 47A10 47A75 47A53 Secondary 47B10 47G10 

Keywords

Factorization of operator-valued analytic functions multiplicity of eigenvalues index computations for finitely meromorphic operator-valued functions 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.Mathematics DepartmentThe University of Tennessee at ChattanoogaChattanoogaUSA

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