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 $\Lambda _{1}(\varepsilon ).$