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A condition for purely absolutely continuous spectrum for CMV operators using the density of states


Authors: Jake Fillman and Darren C. Ong
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 47B36
DOI: https://doi.org/10.1090/proc/13872
Published electronically: October 30, 2017
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Abstract: We prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely purely absolutely continuous if and only if the density of states is absolutely continuous and the Lyapunov exponent vanishes almost everywhere with respect to the same. Both of these results are CMV operator analogues of theorems obtained by Kotani for Schrödinger operators.


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Additional Information

Jake Fillman
Affiliation: Mathematics (MC0123), Virginia Tech, 225 Stanger Street, Blacksburg, Virginia 24061
Email: fillman@vt.edu

Darren C. Ong
Affiliation: Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, 43900 Sepang, Selangor Darul Ehsan, Malaysia
Email: darrenong@xmu.edu.my

DOI: https://doi.org/10.1090/proc/13872
Received by editor(s): December 9, 2016
Published electronically: October 30, 2017
Communicated by: Michael Hitrik
Article copyright: © Copyright 2017 American Mathematical Society