Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem

Golinskii, L.; Kheifets, A.; Peherstorfer, F.; Yuditskii, P.

doi:10.48550/arxiv.0807.4017

Cited by 2 publications

(9 citation statements)

References 11 publications

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“…Theorem 6.12 is an immediate consequence of [6,Theorem 5.3]. The proof of [6,Theorem 5.3] is completely different from the above proof of Theorem 6.12.…”

Section: 26)mentioning

confidence: 96%

“…Theorem 6.12 is an immediate consequence of [6,Theorem 5.3]. The proof of [6,Theorem 5.3] is completely different from the above proof of Theorem 6.12. It is based on a scattering formalism using CMV matrices (For a comprehensive exposition on CMV matrices, we refer the reader to Chapter 4 in the monograph Simon [14].…”

Section: 26)mentioning

confidence: 96%

“…During these discussions they turned our attention to the importance of this description for the solution of the inverse scattering problem (the heart of Faddeev-Marchenko theory), that is, to give necessary and sufficient conditions on a certain class of CMV matrices such that the restriction of this correspondence is one to one (see [6] and the related papers [8] and [11]). Our approach to the description of Helson-Szegő measures differs from the one in [6] in that we investigate this question for CMV matrices in another basis (see [3, Definition 2.2., Theorem 2.13]), namely the one for that CMV matrices have the full GGT representation (see Simon [14, p. 261-262, Remarks and Historical Notes]). However, we consider the study of interrelations between Helson-Szegő measures and Schur parameter sequences to be of particular interest on its own.…”

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confidence: 99%

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Description of Helson-Szegő Measures in Terms of the Schur Parameter Sequences of Associated Schur Functions

Dubovoy

Fritzsche

Kirstein

2012

A Panorama of Modern Operator Theory and Related Topics

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Let be a probability measure on the Borelian -algebra of the unit circle. Then we associate a Schur function in the unit disk with and give characterizations of the case that is a Helson-Szegő measure in terms of the sequence of Schur parameters of . Furthermore, we state some connections of these characterizations with the backward shift. Mathematics Subject Classification (2000). Primary 30E05, 47A57.

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“…Theorem 6.12 is an immediate consequence of [6,Theorem 5.3]. The proof of [6,Theorem 5.3] is completely different from the above proof of Theorem 6.12.…”

Section: 26)mentioning

confidence: 96%

Section: 26)mentioning

confidence: 96%

mentioning

confidence: 99%

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Description of Helson-Szegő Measures in Terms of the Schur Parameter Sequences of Associated Schur Functions

Dubovoy

Fritzsche

Kirstein

2012

A Panorama of Modern Operator Theory and Related Topics

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“…Ya. Kheifets who studied related questions in joint research with F. Peherstorfer and P. M. Yuditskii (see [29], [31], [35]). In Section 8.6 we will comment on some results in [29] which are similar to our Denote by…”

Section: Schur Parameter Sequences Of Polynomial Schur Functions Exam...mentioning

confidence: 99%

“…More specifically, it was shown that a measure µ is a Helson-Szegő measure if and only if some infinite matrix M (which is defined in [29, formulas (4.1) and (4.2)]) generates a bounded operator in ℓ 2 . It was also shown that the boundedness of M is equivalent to the boundedness of another operator matrix L defined in formula (6.4) of [29].…”

Section: This Meansmentioning

confidence: 99%

The S-recurrence of Schur Parameters of Non-inner Rational Schur Functions

Dubovoy¹,

Fritzsche²,

Kirstein³

2010

Topics in Operator Theory

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In [19] there is an approach to the investigation of the pseudocontinuability of Schur functions in terms of Schur parameters. In particular, there was obtained a criterion for the pseudocontinuability of Schur functions and the Schur parameters of rational Schur functions were described. This approach is based on the description in terms of the Schur parameters of the relative position of the largest shift and the largest coshift in a completely nonunitary contraction. It should be mentioned that these results received a further development in [8,[21][22][23][24].This paper is aimed to give a survey about essential results on this direction. The main object in the approach is based on considering a Schur function as characteristic function of a contraction (see Section 1.2). This enables us outgoing from Schur parameters to construct a model of the corresponding contraction (see Section 2). In this model, the relative position of the largest shift and the largest coshift in a completely nonunitary contraction is described in Section 3 and then, based on this model, to find characteristics which are responsible for the pseudocontinuability of Schur functions (see Sections 4 and 5). The further parts of this paper (see Sections 6-8) admit applications of the above results to the study of properties of Schur functions and questions related with them.

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Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem

Cited by 2 publications

References 11 publications

Description of Helson-Szegő Measures in Terms of the Schur Parameter Sequences of Associated Schur Functions

Description of Helson-Szegő Measures in Terms of the Schur Parameter Sequences of Associated Schur Functions

The S-recurrence of Schur Parameters of Non-inner Rational Schur Functions

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