# Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting

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*Joseph A. Ball; Victor Vinnikov*

We present a multivariable setting for Lax-Phillips scattering and for conservative, discrete-time, linear systems. The evolution operator for the Lax-Phillips 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 Cuntz-Toeplitz 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 Cuntz-Toeplitz 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 row-contraction on a Hilbert space.

#### Table of Contents

# Table of Contents

## Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. Functional Models for Row-Isometric/Row-Unitary Operator Tuples 613 free
- Chapter 3. Cuntz Scattering Systems 3542
- Chapter 4. Unitary Colligations 5562
- Chapter 5. Scattering, Systems and Dilation Theory: the Cuntz-Toeplitz Setting 8087
- Bibliography 99106