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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, and Victor Vinnikov, Ben Gurion University of the Negev, Be'er Sheva, Israel
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Memoirs of the American Mathematical Society
2005; 101 pp; softcover
Volume: 178
ISBN-10: 0-8218-3768-0
ISBN-13: 978-0-8218-3768-9
List Price: US$66 Individual Members: US$39.60
Institutional Members: US\$52.80
Order Code: MEMO/178/837

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.