Multiple Geronimus transformations

Derevyagin, Maksym; García-Ardila, Juan C.; Marcellán, Francisco

doi:10.1016/j.laa.2014.04.024

Cited by 29 publications

(31 citation statements)

References 28 publications

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“…However, in the case of the Darboux transformations for CMV, the factor B " A`is not only invertible but also 3-band and upper triangular, so it is an upper H-matrix. Therefore, in this case we can complete the relations (16), (17), (18) with the additional ones…”

Section: Pivq ñ Piiqmentioning

confidence: 99%

“…Higher degree Darboux transformations with parameters for Jacobi matrices have been already considered in the literature [21,28]. Contrary to the case of degree one, they present spurious solutions which are related to Sobolev orthogonal polynomials on the real line [18,19].…”

Section: Higher Degree Transformationsmentioning

confidence: 99%

“…Summarizing, any Hermitian Laurent polynomial ℓ of degree one defines a mapping C ℓ Þ Ñ D between the set C ℓ :" tC CMV : ℓpCq is positive definiteu and the class rSs. This mappping satisfies (16), (17), (18) with A as in (15), and, up to a positive rescaling, any zig-zag bases χ, ω associated with C, D, respectively, are related by…”

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confidence: 99%

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Darboux transformations for CMV matrices

Cantero

Marcellán

Ituarte

et al. 2016

Advances in Mathematics

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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.

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Section: Pivq ñ Piiqmentioning

confidence: 99%

Section: Higher Degree Transformationsmentioning

confidence: 99%

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Darboux transformations for CMV matrices

Cantero

Marcellán

Ituarte

et al. 2016

Advances in Mathematics

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“…Perturbations of a linear functional supported on an infinite subset of the real line (see for example [10,20,24,25,34,50,48]) and the unit circle (see for example [16,17,30,31,35] and the references therein) have been extensively studied in the literature, in particular when these linear functionals are positive definite since they have an integral representation [19,46]. More precisely, there are three perturbations that have historically highlighted, the so called Christoffel [20], Geronimus [34] and Uvarov [48,49] transformations.…”

Section: Introductionmentioning

confidence: 99%

Christoffel Transformation for a Matrix of Bi-variate Measures

García-Ardila

Mañas

Marcellán

2019

Complex Anal. Oper. Theory

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We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms ·, · R and ·, · L P(z 1 ), Q(z 2 ) R = T×T P(z 1 ) † L(z 1 )dµ(z 1 , z 2 )Q(z 2 ),is the set of matrix Laurent polynomials of size p × p and L(z) is a special polynomial in L p×p [z] . A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to ·, · R, (resp. ·, · L) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to dµ(z 1 , z 2 ) is given. (2010). 42C05 15A23 30C10. Keywords. Matrix biorthogonal polynomials, matrix-valued measures, nondegenerate continuous bilinear forms, Gauss-Borel factorization, matrix Christoffel transformations, quasideterminants, block CMV Matrices. Mathematics Subject ClassificationRecently, Cantero, Marcellán, Moral and Velázquez [14] presented an approach to the Darboux transformations for CMV matrices. In particular, for the Christoffel transformation they show that given a Hermitian polynomial L(z), a linear functional µ (supported on the unit circle) and the perturbed oneμ = L(z)µ, if L(C) has Cholesky factorization L(C) = AA † , then L(Ĉ) = A † A, where C andĈ are the CMV matrices associated with µ andμ, respectively.On the other hand, in [38] the authors deal with a measure supported on the unit circle multiplied by a non-negative trigonometric polynomial g(θ). Using the fact that for g(θ) there exists a positive integer number m ∈ N and a polynomial G 2m (z) of degree 2m such that g(θ) = z −m G 2m (z), they give a determinantal expression for the perturbed monic orthogonal polynomial of degree n multiplied by G 2m (z). In this expression, the original orthogonal polynomials, from degree n until degree n + m, their corresponding reversed polynomials (see Eq. (1)) as well as the zeros of the polynomial G 2m (z), are involved.

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“…Uvarov [9] found Christoffel type formulas, and the addition of a finite number of Dirac masses to a linear functional appears in the framework of the spectral analysis of fourth-order linear differential operators with polynomial coefficients and with orthogonal polynomials as eigenfunctions. Geronimus perturbations of degree two of scalar bilinear forms have been very recently treated in [12] and in the general case in [13].…”

Section: Introductionmentioning

confidence: 99%

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

Ariznabarreta¹,

Mañas²,

Toledano³

2018

Stud Appl Math

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Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities such as Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel-Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel-Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e., of the form L(z) = L n z n + · · · + L −n z −n , L n L −n = 0, these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of 2n unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel-Darboux kernels and its mixed versions. Different curves are presented as examples, such as the real line, the circle, the Cassini oval,

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Multiple Geronimus transformations

Cited by 29 publications

References 28 publications

Darboux transformations for CMV matrices

Darboux transformations for CMV matrices

Christoffel Transformation for a Matrix of Bi-variate Measures

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

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