Generalized Fock spaces, interpolation, multipliers, circle geometry

Abstract

By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C which are square integrable with respect to a weight of the type e-Q(z), where Q(z) is a quadratic form such that tr Q > 0. Each such space is in a natural way associated with an (oriented) circle C in C. We consider the problem of interpolation between two Fock spaces. If C0 and C1 are the corresponding circles, one is led to consider the pencil of circles generated by C0 and C1. If H is the one parameter Lie group of Moebius transformations leaving invariant the circles in the pencil, we consider its complexification Hc, which permutes these circles and with the aid of which we can construct the Calderon curve giving the complex interpolation. Similarly, real interpolation leads to a multiplier problem for the transformation that diagonalizes all the operators in Hc. 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.