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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Non-invertibility of certain almost Mathieu operators


Authors: R. Balasubramanian, S. H. Kulkarni and R. Radha
Journal: Proc. Amer. Math. Soc. 129 (2001), 2017-2018
MSC (2000): Primary 47B37; Secondary 15A15
Published electronically: November 30, 2000
MathSciNet review: 1825912
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Abstract:

It is shown that the almost Mathieu operators of the type $Te_n~=$ $e_{n-1}+\lambda sin(2nr)e_n+e_{n+1}$where $\lambda$ is real and $r$ is a rational multiple of $\pi$and $\{e_n:n=1,2,3,...\},$ an orthonormal basis for a Hilbert space, is not invertible.


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Additional Information

R. Balasubramanian
Affiliation: The Institute of Mathematical Sciences, C.I.T. Campus, Madras-600 113, India
Email: balu@imsc.ernet.in

S. H. Kulkarni
Affiliation: Department of Mathematics, Indian Institute of Technology, Madras-600 036, India
Email: shk@acer.iitm.ernet.in

R. Radha
Affiliation: Department of Mathematics, Anna University, Madras-600 025, India
Email: radharam@annauniv.edu, radharam@imsc.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05760-9
PII: S 0002-9939(00)05760-9
Keywords: Almost Mathieu operator, determinant, tridiagonal matrix, tridiagonal operator
Received by editor(s): June 18, 1999
Received by editor(s) in revised form: November 5, 1999
Published electronically: November 30, 2000
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2000 American Mathematical Society