Darboux transformations for CMV matrices

We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measure...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) Vol. 298; pp. 122 - 206
Main Authors: Cantero, M.J., Marcellán, F., Moral, L., Velázquez, L.
Format: Journal Article
Language:English
Published: Elsevier Inc 06.08.2016
Subjects:
Ablowitz–Ladik system
Cholesky factorizations
CMV matrices
Darboux transformations
Measures on the unit circle
Orthogonal Laurent polynomials
Schur flows
Spectral transformations
Darboux transformations
Ablowitz–Ladik system
47B36
15A23
Spectral transformations
CMV matrices
Orthogonal Laurent polynomials
42C05
Cholesky factorizations
Measures on the unit circle
Schur flows
ISSN:0001-8708, 1090-2082
Online Access:Get full text
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Summary:We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations for Jacobi and CMV matrices, like their (almost) isospectrality or the relation that they establish between the corresponding orthogonal polynomials, showing also that both transformations are connected by the Szegő mapping. Nevertheless, we uncover some features of the Darboux transformations for CMV matrices which are in striking contrast with those of the Jacobi case. In particular, when applied to CMV matrices, the matrix realization of the inverse Darboux transformations – what we call ‘Darboux transformations with parameters’ – leads to spurious solutions whose interpretation deserves future research. Such spurious solutions are neither unitary nor band matrices, so Darboux transformations for CMV matrices are much more subject to the subtleties of the algebra of infinite matrices than their Jacobi counterparts. A key role in our theory is played by the Cholesky factorizations of infinite matrices. Actually, the Darboux transformations introduced in this paper are based on the Cholesky factorizations of degree one Hermitian Laurent polynomials evaluated on CMV matrices. These transformations are also generalized to higher degree Laurent polynomials, as well as to the extension of CMV matrices to quasi-definite functionals – what we call ‘quasi-CMV’ matrices. Furthermore, we show that this CMV version of Darboux transformations plays a role in integrable systems like the Schur flows or the Ablowitz–Ladik model which parallels that of Darboux for Jacobi matrices in the Toda lattice.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2016.03.042