On power series subspaces of certain nuclear Fréchet spaces

Abstract

The diametral dimension, \(\Delta (E),\) and the approximate diametral dimension, \(\delta (E)\) of an element E of a class of nuclear Fréchet spaces, which satisfies \((\underline{DN})\) and \(\Omega \) are set theoretically between the respective invariant of power series spaces \(\Lambda _{1}(\varepsilon )\) and \(\Lambda _{\infty }(\varepsilon )\) for some exponent sequence \(\varepsilon .\) Aytuna et al. (Manuscr Math 67:125–142, 1990) proved that E contains a complemented subspace which is isomorphic to \(\Lambda _{\infty }(\varepsilon )\) provided \(\Delta (E)= \Lambda _{\infty }^{\prime }(\varepsilon ))\) and \(\varepsilon \) is stable. In this article, we consider the other extreme case and we prove that, there exist nuclear Fréchet spaces with the properties \((\underline{DN})\) and \(\Omega ,\) even regular nuclear Köthe spaces, satisfying \(\Delta (E)=\Lambda _{1}(\varepsilon )\) such that there is no subspace of E which is isomorphic to \(\Lambda _{1}(\varepsilon ).\)