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In this work we study what we call Siegel--dissipative vector of commutingoperators $(A_1,\ldots, A_{d+1})$ on a Hilbert space $\mathcal H$ and we obtaina von Neumann type inequality which involves the Drury--Arveson space $DA$ onthe Siegel upper half-space $\mathcal U$. The operator $A_{d+1}$ is allowed tobe unbounded and it is the infinitesimal generator of a contraction semigroup$\{e^{-i\tau A_{d+1}}\}_{\tau<0}$. We then study the operator $e^{-i\tauA_{d+1}}A^{\alpha}$ 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 modeloperators on a weighted $L^2$ space. To prove our results we obtain aPaley--Wiener type theorem for $DA$ and we investigate some multiplieroperators on $DA$ as well.

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