Details on the distribution co-orbit space ${\mathcal{H}}_w^{\infty }$

Abstract

Associated with every separable Hilbert space $\mathcal{H}$ and a given localized frame, there exists a natural test function Banach space ${\mathcal{H}}^1$ and a Banach distribution space ${\mathcal{H}}^{\infty }$ so that ${\mathcal{H}}^1\subset \mathcal{H}\subset {\mathcal{H}}^{\infty }$. In this article we close some gaps in the literature and rigorously introduce the space ${\mathcal{H}}^{\infty }$ and its weighted variants ${\mathcal{H}}_w^{\infty }$ in a slightly more general setting and discuss some of their properties. In particular, we compare the underlying weak${}^{\ast }$- with the norm topology associated with ${\mathcal{H}}_w^{\infty }$ and show that $({\mathcal{H}}_w^{\infty },\Vert \cdot {\Vert }_{{\mathcal{H}}_w^{\infty }})$ is a Banach space.