The nuclearity of Gelfand–Shilov spaces and kernel theorems

Abstract

We study the nuclearity of the Gelfand–Shilov spaces ${\mathcal {S}}^{({\mathfrak {M}})}_{({\mathscr {W}})}$ and ${\mathcal {S}}^{\{{\mathfrak {M}}\}}_{\{{\mathscr {W}}\}}$, defined via a weight (multi-)sequence system ${\mathfrak {M}}$ and a weight function system ${\mathscr {W}}$. 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 ${\mathcal {S}}^{(M)}_{(A)}$ and ${\mathcal {S}}^{\{M\}}_{\{A\}}$ (defined via weight sequences M and A) and the Beurling–Björck spaces ${\mathcal {S}}^{(\omega )}_{(\eta )}$ and ${\mathcal {S}}^{\{\omega \}}_{\{\eta \}}$ (defined via weight functions $\omega$ and $\eta$). Our results cover anisotropic cases as well.