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The defect index of singular symmetric linear difference equations with real coefficients


Authors: Guojing Ren and Yuming Shi
Journal: Proc. Amer. Math. Soc. 138 (2010), 2463-2475
MSC (2010): Primary 39A70, 34B20
DOI: https://doi.org/10.1090/S0002-9939-10-10253-6
Published electronically: February 24, 2010
MathSciNet review: 2607876
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Abstract: This paper is concerned with the defect index of singular symmetric linear difference equations of order $ 2n$ with real coefficients and one singular endpoint. We show that their defect index $ d$ satisfies the inequalities $ n\leq d \leq 2n$ and that all values of $ d$ in this range are realized. This parallels the well known result of Glazman for differential equations established about 1950. In addition, several criteria of the limit point and strong limit point cases are established.


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Guojing Ren
Affiliation: School of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China – and – School of Statistics and Mathematics, Shandong Economic University, Jinan, Shandong 250014, People’s Republic of China
Email: rgjmaths@gmail.com

Yuming Shi
Affiliation: School of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
Email: ymshi@sdu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-10-10253-6
Keywords: Singular symmetric linear difference equation, square summable solution, defect index, limit point case, strong limit point case.
Received by editor(s): July 6, 2009
Received by editor(s) in revised form: October 9, 2009
Published electronically: February 24, 2010
Additional Notes: This research was supported by the NNSF of Shandong Province (Grant Y2006A15).
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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