Details on the distribution co-orbit space $\mathcal{H}^{\infty}_w$
Nikolas Hauschka, Peter Balazs, Lukas Köhldorfer
TL;DR
This work rigorously constructs the distribution-type co-orbit space $\mathcal{H}^{\infty}_w$ associated with a localized frame on a separable Hilbert space, and analyzes its topological and Banach-space structure. It defines $\mathcal{H}^{\infty}_w$ via completion in the $\sigma(\mathcal{H},\mathcal{H}^{00})$ topology under uniformly bounded frame-coefficients, and proves that the coefficient operator $C_{\tilde{\psi}}$ is an isometric isomorphism onto a closed subspace of $\ell_w^\infty$, confirming that $\mathcal{H}^{\infty}_w$ is a Banach space. The paper also clarifies the relationship between weak*-type and norm topologies on $\mathcal{H}^{\infty}_w$, and provides a Gramian-based characterization of its image, establishing a solid framework for operator analysis between co-orbit spaces. These results extend the coorbit-space approach to wider settings and yield a robust foundation for applications in time-frequency analysis and quantum physics.
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$^*$- 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.