---
abstract: 'We deduce continuity and Schatten-von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in $(0,\infty ]$. We use these results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate H[ö]{}rmander classes.'
address: 'Department of Mathematics, Linn[æ]{}us University, V[ä]{}xj[ö]{}, Sweden'
author:
- Joachim Toft
title: 'Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or H[ö]{}rmander classes'
---

Introduction {#sec0}
============

The singular values for a linear operator is a non-increasing sequence of non-negative numbers which are strongly linked to questions on continuity and compactness for the operator in the following sense:

-   the operator is continuous, if and only if its singular values are bounded.

-   the operator is compact, if and only if its singular values decay towards zero at infinity. Moreover, fast decays of the singular values permit efficient finite rank approximations.

-   the operator has rank $j\ge 0$, if and only if its singular values of order $j+1$ and higher are zero.

(See [@Ho1; @Gc2] and Section \[sec1\] for definitions.)

In particular, there is a strong connection between the decay of the singular values and finding 