Integral Equations and Operator Theory

, Volume 16, Issue 2, pp 244–266

For which reproducing kernel Hilbert spaces is Pick's theorem true?

  • Peter Quiggin
Article

DOI: 10.1007/BF01358955

Cite this article as:
Quiggin, P. Integr equ oper theory (1993) 16: 244. doi:10.1007/BF01358955

Abstract

Pick's theorem tells us that there exists a function inH, which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH2, and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.

MSC 1991

Primary 30E05 Secondary 46E22 47A20 

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Peter Quiggin
    • 1
  1. 1.Mathematics DepartmentLancaster UniversityLancasterUK

Personalised recommendations