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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
Available Formats:
Electronic ISBN: 978-1-4704-0438-3
Product Code: MEMO/178/837.E
List Price: $66.00 MAA Member Price:$59.40
AMS Member Price: $39.60 Click above image for expanded view 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  List Price:$66.00 MAA Member Price: $59.40 AMS Member Price:$39.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1782005; 101 pp
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.

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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Volume: 1782005; 101 pp
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.