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Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting
 
Joseph A. Ball Virginia Polytechnic Institute and State University, Blacksburg, VA
Victor Vinnikov Ben Gurion University of the Negev, Be’er Sheva, Israel
Front Cover for Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting
Available Formats:
Electronic ISBN: 978-1-4704-0438-3
Product Code: MEMO/178/837.E
101 pp 
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
Front Cover for Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting
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  • Front Cover for Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting
  • Back Cover for Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting
Lax-Phillips Scattering and Conservative Linear Systems: A Cuntz-Algebra Multidimensional Setting
Joseph A. Ball Virginia Polytechnic Institute and State University, Blacksburg, VA
Victor Vinnikov Ben Gurion University of the Negev, Be’er Sheva, Israel
Available Formats:
Electronic ISBN:  978-1-4704-0438-3
Product Code:  MEMO/178/837.E
101 pp 
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1782005
    MSC: Primary 47; Secondary 13; 93;

    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.

    Readership

    Graduate students and research mathematicians interested in analysis.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Functional models for row-isometric/row-unitary operator tuples
    • 3. Cuntz scattering systems
    • 4. Unitary colligations
    • 5. Scattering, systems and dilation theory: the Cuntz-Toeplitz Setting
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Volume: 1782005
MSC: Primary 47; Secondary 13; 93;

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.

Readership

Graduate students and research mathematicians interested in analysis.

  • Chapters
  • 1. Introduction
  • 2. Functional models for row-isometric/row-unitary operator tuples
  • 3. Cuntz scattering systems
  • 4. Unitary colligations
  • 5. Scattering, systems and dilation theory: the Cuntz-Toeplitz Setting
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