Abstract
In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.
Dedicated to our mentor and friend Rien Kaashoek on the occasion of his eightieth birthday.
This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 90670 and 93406).
Part of the research was done during a sabbatical of the third author, in which time several research visits to VU Amsterdam and North-West University were made. Support from University of Cape Town and the Department of Mathematics, VU Amsterdam is gratefully acknowledged.
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Groenewald, G.J., ter Horst, S., Jaftha, J., Ran, A.C.M. (2018). A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties. In: Bart, H., ter Horst, S., Ran, A., Woerdeman, H. (eds) Operator Theory, Analysis and the State Space Approach. Operator Theory: Advances and Applications, vol 271. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04269-1_10
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DOI: https://doi.org/10.1007/978-3-030-04269-1_10
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-04268-4
Online ISBN: 978-3-030-04269-1
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