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 $$(A_1,\ldots , A_{d+1})$$ ( A 1 , … , A d + 1 ) on a Hilbert space $${{\mathcal {H}}}$$ H and we obtain a von Neumann type inequality which involves the Drury–Arveson space DA on the Siegel upper half-space $${{\mathcal {U}}}$$ U . The operator $$A_{d+1}$$ A d + 1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $$\{e^{-i\tau A_{d+1}}\}_{\tau <0}$$ { e - i τ A d + 1 } τ < 0 . We then study the operator $$e^{-i\tau A_{d+1}}A^{\alpha }$$ e - i τ A d + 1 A α where $$A^{\alpha }=A_1^{\alpha _1}\cdots A^{\alpha _d}_d$$ A α = A 1 α 1 ⋯ A d α d for $$\alpha \in {\mathbb N}_0^d$$ α ∈ N 0 d and prove that can be studied by means of model operators on a weighted $$L^2$$ 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.

∗-Controlled frames in Hilbert C∗-modules

The main purpose of this paper is to introduce the notion of ∗-T-controlled frames in Hilbert [Formula: see text]-modules. We present some results of frames in the view of ∗-T-controlled frames in Hilbert [Formula: see text]-modules. Also, we define ∗-w-frames and give their relations to ∗-T-controlled frames and multiplier operators.

Smoothness of functions versus smoothness of approximation processes

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.