Generalized Fock spaces, interpolation, multipliers, circle geometry (Q1914390)

Summary: 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 \(\text{tr } Q>0\). Each such space is in a natural way associated with an (oriented) circle \({\mathcal C}\) in \(\mathbb{C}\). We consider the problem of interpolation between two Fock spaces. If \({\mathcal C}_0\) and \({\mathcal C}_1\) are the corresponding circles, one is led to consider the pencil of circles generated by \({\mathcal C}_0\) and \({\mathcal C}_1\). 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 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.