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,… , A_d+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 A_d+1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup {e^-iτ A_d+1}_τ <0 . We then study the operator e^-iτ A_d+1A^α where A^α=A_1^α _1⋯ A^α _d_d for α∈ℕ_0^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.