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

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping $\mathcal{S}({\mathbb{R}}^n)$ into ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$, 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.