Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables
✍ Scribed by Joseph A Ball; Tavan T Trent
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 669 KB
- Volume
- 157
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
Recently J. Agler studied the class S d of scalar-valued, analytic functions of d complex variables f for which f (T 1 , ..., T d ) has norm at most 1 for any collection of d commuting contractions (T 1 , ..., T d ) on a Hilbert space H. Among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d-variable linear system, as well as a Nevanlinna Pick interpolation theorem for this class of functions. In this note we examine the system theory aspects and uniqueness of the transfer function representation, and give a simpler proof of the Nevanlinna Pick interpolation theorem for the class S d and obtain a d-variable version of the Toeplitz corona theorem. By using ideas of Arov and Grossman introduced for 1-variable problems, as a bonus we obtain a collection of linear fractional maps which parametrize the set of all S d solutions of an interpolation problem.