Compactness Criteria for Sets and Operators in the Setting of Continuous Frames

TL;DR

This paper develops criteria for relative compactness of sets and operators in the context of continuous frames and coorbit theory, with applications to magnetic pseudodifferential calculus.

Contribution

It introduces new compactness criteria involving tightness properties in continuous frame settings, including a magnetic pseudodifferential calculus extension.

Findings

Criteria for relative compactness of sets and operators

Application to magnetic pseudodifferential calculus

Development of a symbolic calculus for continuous frames

Abstract

To a generalized tight continuous frame in a Hilbert space $\H$ indexed by a locally compact space $\Si$ endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on $\Si$ in spaces of vectors comparable with $\H$. If the continuous frame is provided by the action of a suitable family of bounded operators on a fixed window, a symbolic calculus emerges, assigning operators in $\H$ to functions on $\Si$. We give some criteria of relative compactness for sets and for families of compact operators, involving tightness properties in terms of objects canonically associated to the frame. Particular attention is dedicated to a magnetic version of the pseudodifferential calculus.