Electronic ISBN:  9781470404383 
Product Code:  MEMO/178/837.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 178; 2005; 101 ppMSC: Primary 47; Secondary 13; 93;
We present a multivariable setting for LaxPhillips scattering and for conservative, discretetime, linear systems. The evolution operator for the LaxPhillips scattering system is an isometric representation of the Cuntz algebra, while the nonnegative time axis for the conservative, linear system is the free semigroup on \(d\) letters. The correspondence between scattering and system theory and the roles of the scattering function for the scattering system and the transfer function for the linear system are highlighted. Another issue addressed is the extension of a given representation of the CuntzToeplitz algebra (i.e., a row isometry) to a representation of the Cuntz algebra (i.e., a row unitary); the solution to this problem relies on an extension of the Szegö factorization theorem for positive Toeplitz operators to the CuntzToeplitz algebra setting. As an application, we obtain a complete set of unitary invariants (the characteristic function together with a choice of “Haplitz” extension of the characteristic function defect) for a rowcontraction on a Hilbert space.
ReadershipGraduate students and research mathematicians interested in analysis.

Table of Contents

Chapters

1. Introduction

2. Functional models for rowisometric/rowunitary operator tuples

3. Cuntz scattering systems

4. Unitary colligations

5. Scattering, systems and dilation theory: the CuntzToeplitz Setting


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We present a multivariable setting for LaxPhillips scattering and for conservative, discretetime, linear systems. The evolution operator for the LaxPhillips scattering system is an isometric representation of the Cuntz algebra, while the nonnegative time axis for the conservative, linear system is the free semigroup on \(d\) letters. The correspondence between scattering and system theory and the roles of the scattering function for the scattering system and the transfer function for the linear system are highlighted. Another issue addressed is the extension of a given representation of the CuntzToeplitz algebra (i.e., a row isometry) to a representation of the Cuntz algebra (i.e., a row unitary); the solution to this problem relies on an extension of the Szegö factorization theorem for positive Toeplitz operators to the CuntzToeplitz algebra setting. As an application, we obtain a complete set of unitary invariants (the characteristic function together with a choice of “Haplitz” extension of the characteristic function defect) for a rowcontraction on a Hilbert space.
Graduate students and research mathematicians interested in analysis.

Chapters

1. Introduction

2. Functional models for rowisometric/rowunitary operator tuples

3. Cuntz scattering systems

4. Unitary colligations

5. Scattering, systems and dilation theory: the CuntzToeplitz Setting