Fock Spaces Corresponding to Positive Definite 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 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. Introduction The classical Segal-Bargmann transform is an integral transform which defines a unitary isomorphism of L(R) onto the Hilbert space F(C) of entire functions on C which are square integrable with respect to the Gaussian measure μ = π−ne−|z| 2 dxdy, where dxdy stands for the Lebesgue measure on R ≃ C, see [1, 3, 4, 5, 10, 11]. There have been several generalizations of this transform, based on the heat equation or the representation theory of Lie groups [6, 9, 12]. In particular, it was shown in [9] that the Segal-Bargmann transform is a special case of the restriction principle, i.e., construction of unitary isomorphisms based on the polarization of a restriction map. This principle was first introduced in [9], see also [8], where several examples were explained from that point of view. In short the restriction principle can be explained in the following way. Let MC be a complex manifold and let M ⊂ MC be a totally real submanifold. Let F = F(MC) be a Hilbert space of holomorphic functions on MC such that the evaluation maps F ∋ F 7→ F (z) ∈ C are continuous for all z ∈ MC, i.e., F is a reproducing Hilbert space. There exists a function K : MC ×MC → C holomorphic in the first variable, anti-holomorphic in the second variable, and such that the following hold: (a) K(z, w) = K(w, z) for all z, w ∈ MC; (b) If Kw(z) := K(z, w) then Kw ∈ F and F (w) = (F,Kw), ∀F ∈ F, z ∈ MC . The function K is the reproducing kernel for the Hilbert space. Let D : M → C∗ be measurable. Then the restriction map RF := DF |M is injective. Assume that there is a measure μ on M such that RF ∈ L(M,μ) for all F in a dense subset of 2000 Mathematics Subject Classification. Primary 46E22, Secondary 81S10.