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 $\mathbb{C}$ 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 $\mathbb{C}$. We consider the problem of interpolation betweeumn two Fock spaces. If $C_0$ and $C_1$ are the corresponding circles, one is led to consider the pencil of circles generated by $C_0$ and $C_1$. If $H$ is the one parameter Lie group of Moebius transformations leaving invariant the circles in the pencil, we consider its complexification $H^c$ 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 $H^c$. 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.