The Drury--Arveson space on the Siegel upper half-space and a von Neumann type inequality

In this work we study what we call Siegel--dissipative vector of commuting operators $(A_1,\ldots, A_{d+1})$ on a Hilbert space $\mathcal H$ and we obtain a von Neumann type inequality which involves the Drury--Arveson space $DA$ on the Siegel upper half-space $\mathcal U$. The operator $A_{d+1}$ is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $\{e^{-i\tau A_{d+1}}\}_{\tau<0}$. We then study the operator $e^{-i\tau A_{d+1}}A^{\alpha}$ where $A^{\alpha}=A_1^{\alpha_1}\cdots A^{\alpha_d}_d$ for $\alpha\in\mathbb N^d_0$ and prove that can be studied by means of model operators on a weighted $L^2$ space. To prove our results we obtain a Paley--Wiener type theorem for $DA$ and we investigate some multiplier operators on $DA$ as well.