Fock spaces corresponding to positive definite linear transformations

2003/04/23 by Raymond C. Fabec, Fabec, R., Gestur Ólafsson +3
Mathematics · #46E22 #81S10 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.math/0304358

openalex publication_date 2003/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 √ det A πⁿe^-\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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