Holomorphic Fock spaces for positive linear transformations
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 dμA(z) = √detA/πn e-Re(Az,z) dz is described in terms of the linear and antilinear 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.
Publication Source (Journal or Book title)
Fabec, R., Ólafsson, G., & Sengupta, A. (2006). Holomorphic Fock spaces for positive linear transformations. Mathematica Scandinavica, 98 (2), 262-281. https://doi.org/10.7146/math.scand.a-14995