Integral Equations and Operator Theory

, Volume 56, Issue 2, pp 229–256 | Cite as

The Spectral Carathéodory-Fejér Problem

  • H. -N. Huang
  • S. A. M. Marcantognini
  • N. J. Young
Article

Abstract.

The problem of the title is to construct an analytic k  ×   k matrix-valued function in the unit disc with a number of prescribed derivatives at 0 and with spectral radius bounded by 1. We show that the problem can be reduced to an interpolation problem for the symmetrized polydisc \({\mathbb{G}_{k}}\), and thereby show that, in the case of derivatives of orders 0 and 1 being prescribed, the problem is equivalent to the infinitesimal Kobayashi extremal problem for \({\mathbb{G}_{k}}\), which is solved completely in the case k = 2.

Mathematics Subject Classification (2000).

Primary 30E05 47A56 Secondary 32F45 93B50 

Keywords.

Spectral Carathéodory-Fejér interpolation Schur functions spectral Nevanlinna-Pick interpolation theory symmetrized polydisc Kobayashi metric 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  • H. -N. Huang
    • 1
  • S. A. M. Marcantognini
    • 2
  • N. J. Young
    • 3
  1. 1.Department of MathematicsTunghai UniversityTaichungTaiwan
  2. 2.Department of MathematicsInstituto Venezolano de Investigaciones CientíficasCaracasVenezuela
  3. 3.School of Mathematics and StatisticsUniversity of Newcastle upon TyneEngland

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