Composition operators on the Schwartz space

TL;DR

This paper investigates the properties of composition operators on the Schwartz space, establishing their non-compactness and providing conditions for their range to be closed, based on multipliers and smooth function spaces.

Contribution

It offers new criteria for the closed range property of composition operators on the Schwartz space, linking it to multipliers and smooth function spaces.

Findings

Composition operators are never compact on the Schwartz space.

Necessary and sufficient conditions for the range to be closed are identified.

Conditions are expressed via multipliers and smooth function space properties.

Abstract

We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition operator to be closed. These conditions are expressed in terms of multipliers for the Schwartz class and the closed range property of the corresponding operator considered in the space of smooth functions.