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

Abstract

In this work we study what we call Siegel--dissipative vector of commuting operators $(A_1,\dots ,A_{d+1})$ 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 $A^{\alpha }=A_1^{{\alpha }_1}\cdots A_d^{{\alpha }_d}$ for $\alpha \in {\mathbb{N}}_0^d$ 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.

17 pages. Typos were fixed and the references updated

References in corpus (1)

• Holomorphic Function Spaces on Homogeneous Siegel Domains

Cited by in corpus (1)

• Carleson and reverse Carleson measures on homogeneous Siegel domains