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

Joseph A. Ball, Victor Vinnikov

Research output: Contribution to journalReview articlepeer-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
JournalMemoirs of the American Mathematical Society
Volume178
Issue number837
DOIs
StatePublished - 1 Jan 2005

Keywords

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

Fingerprint

Dive into the research topics of 'Lax-Phillips scattering and conservative linear systems: A Cuntz-algebra multidimensional setting'. Together they form a unique fingerprint.

Cite this