  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
$\alpha\in\mathbb N^d$ 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.
