Generalised Fock spaces, abstract 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 > O. 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 t.he circles in the pencil, we consider its complexification H^c. which permutes these circles and with the aid of which we call 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 H^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.