By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane which are square integrable with respect to a weight of the type , where is a quadratic form such that tr. Each such space is in a natural way associated with an (oriented) circle in . We consider the problem of interpolation betweeumn two Fock spaces. If and are the corresponding circles, one is led to consider the pencil of circles generated by and . If is the one parameter Lie group of Moebius transformations leaving invariant the circles in the pencil, we consider its complexification 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 . 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.
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Jaak Peetre, Sundaram Thangavelu, Nils-Olof Wallin, Generalized Fock spaces, interpolation, multipliers, circle geometry. Rev. Mat. Iberoam. 12 (1996), no. 1, pp. 63–110DOI 10.4171/RMI/195