About the nuclearity of ${\mathcal S}_{(M_{p})}$ and ${\mathcal S}_{\omega}$

TL;DR

This paper establishes sufficient conditions under which certain sequence and weighted function spaces are nuclear, using an isomorphism to relate these spaces and analyzing their properties.

Contribution

It provides new criteria for the nuclearity of the spaces ${\mathcal S}_{(M_p)}$ and ${\mathcal S}_\omega$, extending previous results through an isomorphism approach.

Findings

${\mathcal S}_{(M_p)}$ is nuclear under certain conditions

${\mathcal S}_\omega$ is nuclear if the weight function satisfies a mild condition

The paper links sequence spaces with weighted generalized function spaces

Abstract

We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space ${\mathcal S}_{(M_p)}$ to be nuclear. As a consequence, we obtain that for a weight function $\omega$ satisfying the mild condition: $2\omega(t)\leq \omega(Ht)+H$ for some $H>1$ and for all $t\geq0$, the space ${\mathcal S}_\omega$ in the sense of Bj\"orck is also nuclear.