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 (A1,...,Ad+1) on a Hilbert space H and we obtain a von Neumann type inequality which involves the Drury–Arveson space DA on the Siegel upper half-space U. The operator Ad+1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup {e−iτAd+1 }τ<0. We then study the operator e−iτAd+1Aα where Aα = Aα1 1 ··· Aαd d for α ∈ Nd 0 and prove that can be studied by means of model operators on a weighted L2 space. To prove our results we obtain a Paley–Wiener type theorem for DA and we investigate some multiplier operators on DA as well.