Lax-Phillips scattering and conservative linear systems: a Cuntz-algebra multidimensional setting

Joseph A. Ball, Victor Vinnikov

Research output: Book/ReportBookpeer-review

42 Scopus citations

Abstract

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.

Original languageEnglish
Place of PublicationProvidence, RI
PublisherAmerican Mathematical Society
Number of pages101
Volume178
ISBN (Electronic)0821837680, 1470404389, 9780821837689
DOIs
StatePublished - 1 Jan 2005

Publication series

NameMemoirs of the American Mathematical Society
PublisherAmerican Mathematical Society
ISSN (Print)0065-9266

Keywords

  • Formal power series
  • Functional model
  • Incoming and outgoing space
  • Noncommuting indeterminants
  • Row contraction
  • Row isometry
  • Scattering function
  • Unitary colligation

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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