On Power Series Subspaces of Certain Nuclear Frechet Spaces

The diametral dimension, $\mathrm{\Delta }(E)$, and the approximate diametral dimension, $\delta (E)$ of an element $E$ of a large class of nuclear Fréchet spaces are set theoretically between the corresponding invariant of power series spaces ${\mathrm{\Lambda }}_1(\epsilon )$ and ${\mathrm{\Lambda }}_{\mathrm{\infty }}(\epsilon )$ for some exponent sequence $\epsilon$. Aytuna et al., \cite{AKT2}, proved that $E$ contains a complemented subspace which is isomorphic to ${\mathrm{\Lambda }}_{\mathrm{\infty }}(\epsilon )$ provided $\mathrm{\Delta }(E)=\mathrm{\Delta }({\mathrm{\Lambda }}_{\mathrm{\infty }}(\epsilon ))$ and $\epsilon$ is stable. In this article, we will consider the other extreme case and we proved that in this large family, there exist nuclear Fréchet spaces, even regular nuclear Köthe spaces, satisfying $\mathrm{\Delta }(E)=\mathrm{\Delta }({\mathrm{\Lambda }}_1(\epsilon ))$ such that there is no subspace of $E$ which is isomorphic to ${\mathrm{\Lambda }}_1(\epsilon )$.