Unbounded Operators on the Segal–Bargmann Space

In this paper we analyse the domains of differential and multiplication operators on the Segal–Bargmann space. We consider the basic estimate for the \(\partial \) - complex, remark that this estimate is closely related to the uncertainty principle in quantum mechanics and compute the Bergman kernel of the graph norm. It is shown that the set of all functions u in the Segal–Bargmann \(A^2(\mathbb C, e^{-|z|^2})\) such that the multiplication with a polynomial p is norm bounded gives a relatively compact subset of the Segal–Bargmann space. In the following section we give a survey of recent results on the \(\partial \) -complex on weighted Bergman spaces on Hermitian manifolds, analysing metrics which produce a similar duality between differentiation and multiplication as in the Segal–Bargmann space. Finally we study the basic estimates and the corresponding questions of compactness for the generalized \(\partial \) -complex.