Integral Equations and Operator Theory

, Volume 88, Issue 4, pp 501–534 | Cite as

The Characteristic Function for Complex Doubly Infinite Jacobi Matrices

Open Access
Article

Abstract

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.

Keywords

Doubly infinite Jacobi matrix Spectral analysis Characteristic function 

Mathematics Subject Classification

Primary 47B36 Secondary 15A18 

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Information TechnologyCzech Technical University in PraguePragueCzech Republic
  2. 2.Department of MathematicsStockholm UniversityStockholmSweden

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