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Fock spaces corresponding to positive definite linear transformations

R. Fabec·G. Olafsson·A. Sengupta
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Abstract

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $dμ_A(z)=\frac {\sqrt {\det A}} {π^n}e^{-{\rm Re}< Az,z>} dz$ is described in terms of the holomorphic--antiholomorphic decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal--Bargmann transform, which we also study in the Gaussian-measure setting.

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R. Fabec, G. Olafsson, A. Sengupta. 2003-04-23. Fock spaces corresponding to positive definite linear transformations. https://arxiv.org/abs/math/0304358

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