Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators (Q1304743)

Let \(L^2(\mu)\) denote the Hilbert space of all complex-valued square integrable functions \(f\) on complex \(n\)-space \(\mathbb{C}^n\) with respect to the Gaussian measure \(\mu\) on \(\mathbb{C}^n\), and let \(B\subset L^2(\mu)\) be the subspace of entire functions. The author generalizes the Newman-Shapiro isometry theorem for \(B\) by letting the range of \(f\) be \(\mathbb{C}^m\) and later a separable Hilbert space. He then studies the action of Toeplitz operators on \(f\) and gives several examples. These are followed by various technical results on extensions of Toeplitz operators, density of domains, cores, closedness, and boundedness from below.