The nuclearity of Gelfand-Shilov spaces and kernel theorems

Abstract

We study the nuclearity of the Gelfand-Shilov spaces ${\mathcal{S}}_{(\mathcal{W})}^{(\mathfrak{M})}$ and ${\mathcal{S}}_{\{\mathcal{W}\}}^{\{\mathfrak{M}\}}$, defined via a weight (multi-)sequence system $\mathfrak{M}$ and a weight function system $\mathcal{W}$. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for Köthe sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces ${\mathcal{S}}_{(A)}^{(M)}$ and ${\mathcal{S}}_{\{A\}}^{\{M\}}$ (defined via weight sequences $M$ and $A$) and the Beurling-Björck spaces ${\mathcal{S}}_{(\eta )}^{(\omega )}$ and ${\mathcal{S}}_{\{\eta \}}^{\{\omega \}}$ (defined via weight functions $\omega$ and $\eta$). Our results cover anisotropic cases as well.

24 pages

References in corpus (4)

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