On Power Series Subspaces of Certain Nuclear Frechet Spaces

Abstract

The diametral dimension, $\Delta(E)$, and the approximate diametral dimension, $\delta(E)$ of an element $E$ of a large class of nuclear Fr\'echet spaces are set theoretically between the corresponding invariant of power series spaces $\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}(\varepsilon)$ for some exponent sequence $\varepsilon$. Aytuna et al., \cite{AKT2}, proved that $E$ contains a complemented subspace which is isomorphic to $\Lambda_{\infty}(\varepsilon)$ provided $\Delta(E)=\Delta( \Lambda_{\infty}(\varepsilon))$ and $\varepsilon$ is stable. In this article, we will consider the other extreme case and we proved that in this large family, there exist nuclear Fr\'echet spaces, even regular nuclear K\"othe spaces, satisfying $\Delta(E)=\Delta(\Lambda_{1}(\varepsilon))$ such that there is no subspace of $E$ which is isomorphic to $\Lambda_{1}(\varepsilon)$.