# On the anti-Wick symbol as a Gelfand-Shilov generalized function

## TL;DR

This paper demonstrates that the anti-Wick symbol of certain operators can be rigorously defined within the Gelfand-Shilov generalized function framework, expanding the understanding of symbol representations beyond tempered distributions.

## Contribution

It proves that anti-Wick symbols can be characterized as Gelfand-Shilov generalized functions, even when they are not tempered distributions, using specific space embeddings.

## Key findings

- Anti-Wick symbols can be defined as Gelfand-Shilov generalized functions.
- The work establishes embeddings between Schwartz, Gelfand-Shilov, and Gevrey spaces.
- Results extend the class of symbols representable in generalized function frameworks.

## Abstract

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping $ {\cal S}(\R^n)$ into ${\cal S}'(\R^n)$, which is generally not a tempered distribution, can still be defined as a Gelfand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gelfand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.