The nuclearity of Gelfand–Shilov spaces and kernel theorems

• Autores: Andreas Debrouwere, Lenny Neyt, Jasson Vindas
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 1, 2021, págs. 203-227
• Idioma: inglés
• DOI: 10.1007/s13348-020-00286-2
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• Resumen

  • 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.