Duality in Segal–Bargmann spaces

Abstract: For , the Bargmann projection is the orthogonal projection from onto the holomorphic subspace , where is the standard Gaussian probability measure on with variance . The space is classically known as the Segal–Bargmann space. We show that extends to a bounded operator on , and calculate the exact norm of this scaled Bargmann projection. We use this to show that the dual space of the -Segal–Bargmann space is an Segal–Bargmann space, but with the Gaussian measure scaled differently: (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms. [Copyright &y& Elsevier]