Generalized Fock spaces, interpolation, multipliers, circle geometry

  • Jaak Peetre

    Lund University, Sweden
  • Sundaram Thangavelu

    Indian Institute of Science, Bangalore, India
  • Nils-Olof Wallin

    Lund University, Sweden


By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C\mathbb C which are square integrable with respect to a weight of the type eQ(z)e^{–Q(z)}, where Q(z)Q(z) is a quadratic form such that trQ>0Q>0. Each such space is in a natural way associated with an (oriented) circle CC in C\mathbb C. We consider the problem of interpolation betweeumn two Fock spaces. If C0C_0 and C1C_1 are the corresponding circles, one is led to consider the pencil of circles generated by C0C_0 and C1C_1. If HH is the one parameter Lie group of Moebius transformations leaving invariant the circles in the pencil, we consider its complexification HcH^c which permutes these circles and with the aid of which we can construct the "Calderón curve" giving the complex interpolation. Similarly, real interpolation leads to a multiplier problem for the transforrnation that diagonalizes all the operators in HcH^c. It turns out that the result is rather sensitive to the nature of the pencil, and we obtain nearly complete results for elliptic and parabolic pencils only.

Cite this article

Jaak Peetre, Sundaram Thangavelu, Nils-Olof Wallin, Generalized Fock spaces, interpolation, multipliers, circle geometry. Rev. Mat. Iberoam. 12 (1996), no. 1, pp. 63–110

DOI 10.4171/RMI/195