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Pick Matrices for Schur Multipliers

  • Harry Dym
  • Dan Volok
Chapter
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Part of the Operator Theory: Advances and Applications book series (OT, volume 197)

Abstract

It is well known that the classical Nevanlinna-Pick problem for holomorphic contractive functions in the open unit disk is solvable if and only if a matrix P with entries of the form
$$ p_{jk} = \frac{{1 - \overline {\eta _j } \eta _k }} {{1 - \omega _k \overline {\omega _j } }}, j.k = 1, \ldots ,n, $$
, that is based on the data of the problem is positive semidefinite. The purpose of this purely expository note is to draw attention to another matrix that arises in the theory of interpolation problems for multipliers which deserves to be better known. This matrix and other more general forms are discussed in [AlB97] and [AlBL96]. Our interest in this problem was aroused by the formula
$$ p_{jk} = \frac{{1 - \overline {\eta _j } (1 - \omega _j^\ell \overline {\omega _k } ^\ell )\eta _k }} {{1 - \omega _j \overline {\omega _k } }}, j.k = 1, \ldots ,n, $$
that was mentioned by M.A. Kaashoek in a lecture at the IWOTA conference in Blacksburg, Virginia, as a byproduct of his joint investigations with C. Foias and A.E. Frazho [FFK02] into constrained lifting problems.

Mathematics Subject Classification (2000)

Primary 30E05 Secondary 47B32 47B38 46E22 

Keywords

Tangential interpolation Schur class multipliers Pick matrices reproducing kernel Hilbert spaces 

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References

  1. [AlB97] D. Alpay and V. Bolotnikov, On tangential interpolation in reproducing kernel Hilbert modules and applications; in Topics in Interpolation Theory (Leipzig, 1994) (H. Dym, B. Fritzsche, V. Katsnelson and B. Kirstein, eds.), Oper. Theory Adv. Appl. 95, Birkhäuser, Basel, 1997, pp. 37–68.Google Scholar
  2. [AlBL96] D. Alpay, V. Bolotnikov and P. Loubaton, On tangential H 2 interpolation with second order norm constraints. Integ. Equat. Oper. Th. 24 (1996), no. 2, 156–178.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Dy89] H. Dym, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. CBMS Regional Conference Series in Mathematics, 71, Amer. Math. Soc., Providence, RI, 1989.Google Scholar
  4. [Dy94] H. Dym, Expository Review of The Commutant Lifting Approach to Interpolation by C. Foias and A.E. Frazho, Bull. Amer. Math. Soc. 31 (1994), no. 1, 125–140.CrossRefMathSciNetGoogle Scholar
  5. [Dy03] H. Dym, Riccati equations and bitangential interpolation problems with singular Pick matrices; in Fast Algorithms for Structured Matrices: Theory and Applications (South Hadley, MA, 2001) (V. Olshevsky, ed.), Contemp. Math. 323, Amer. Math. Soc., Providence, RI, 2003, pp. 361–391.Google Scholar
  6. [FFK02] C. Foias, A. Frazho and M.A. Kaashoek Relaxation of metric constrained interpolation and a new lifting theorem. Integ. Equat. Oper. Th. 42 (2002), no. 3, 253–310.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Harry Dym
    • 1
  • Dan Volok
    • 2
  1. 1.Department of MathematicsThe Weizmann InstituteRehovotIsrael
  2. 2.Department of MathematicsKansas State UniversityManhattanUSA

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