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Finite-Gap CMV Matrices : Periodic Coordinates and a Magic Formula

Christiansen, Jacob S. LU ; Eichinger, Benjamin LU and Vandenboom, Tom (2021) In International Mathematics Research Notices 2021(18). p.14016-14085
Abstract

We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum E and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as E-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them "MCMV."Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated... (More)

We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum E and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as E-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them "MCMV."Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.

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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
International Mathematics Research Notices
volume
2021
issue
18
pages
70 pages
publisher
Oxford University Press
external identifiers
  • scopus:85122299255
ISSN
1073-7928
DOI
10.1093/imrn/rnz213
language
English
LU publication?
yes
additional info
Publisher Copyright: © 2020 The Author(s). Published by Oxford University Press. All rights reserved.
id
109abbdf-97a5-426d-956a-16c92aa06402
date added to LUP
2022-02-21 14:51:53
date last changed
2025-10-14 09:18:00
@article{109abbdf-97a5-426d-956a-16c92aa06402,
  abstract     = {{<p>We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum E and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as E-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them "MCMV."Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities. </p>}},
  author       = {{Christiansen, Jacob S. and Eichinger, Benjamin and Vandenboom, Tom}},
  issn         = {{1073-7928}},
  language     = {{eng}},
  month        = {{09}},
  number       = {{18}},
  pages        = {{14016--14085}},
  publisher    = {{Oxford University Press}},
  series       = {{International Mathematics Research Notices}},
  title        = {{Finite-Gap CMV Matrices : Periodic Coordinates and a Magic Formula}},
  url          = {{http://dx.doi.org/10.1093/imrn/rnz213}},
  doi          = {{10.1093/imrn/rnz213}},
  volume       = {{2021}},
  year         = {{2021}},
}