Schatten properties, nuclearity and minimality of shift invariant spaces

TL;DR

This paper extends minimality properties of shift-invariant Banach spaces to quasi-Banach spaces and applies these results to characterize Schatten and nuclear operators between modulation spaces.

Contribution

It generalizes Feichtinger's minimality property to quasi-Banach spaces and uses this to analyze Schatten and nuclearity of certain operators on modulation spaces.

Findings

Proves $ ext{Op}(a)$ is Schatten-$q$ from $M^$ to $M^p$

Shows $ ext{Op}(a)$ is $r$-nuclear from $M^$ to $M^r$ when $a  M^r$

Extends minimality properties to matrix classes and quasi-Banach spaces

Abstract

We extend Feichtinger's minimality property on smallest non-trivial time-frequency shift invariant Banach spaces, to the quasi-Banach case. Analogous properties are deduced for certain matrix classes. We use these results to prove that $\operatorname{Op}(a)$ is a Schatten-$q$ operator from $M^\infty$ to $M^p$ and $r$-nuclear operator from $M^\infty$ to $M^r$ when $a\in M^r$ for suitable $p$, $q$ and $r$ in $(0,\infty ]$.