On power series subspaces of certain nuclear Fréchet spaces

The diametral dimension, \(\Delta (E),\) $\mathrm{\Delta }(E),$ and the approximate diametral dimension, \(\delta (E)\) $\delta (E)$ of an element E of a class of nuclear Fréchet spaces, which satisfies \((\underline{DN})\) $(\underline{\mathit{DN}})$ and \(\Omega \) $\mathrm{\Omega }$ are set theoretically between the respective invariant of power series spaces \(\Lambda _{1}(\varepsilon )\) ${\mathrm{\Lambda }}_1(\epsilon )$ and \(\Lambda _{\infty }(\varepsilon )\) ${\mathrm{\Lambda }}_{\infty }(\epsilon )$ for some exponent sequence \(\varepsilon .\) $\epsilon .$ Aytuna et al. (Manuscr Math 67:125–142, 1990) proved that E contains a complemented subspace which is isomorphic to \(\Lambda _{\infty }(\varepsilon )\) ${\mathrm{\Lambda }}_{\infty }(\epsilon )$ provided \(\Delta (E)= \Lambda _{\infty }^{\prime }(\varepsilon ))\) $\mathrm{\Delta }(E)={\mathrm{\Lambda }}_{\infty }^{\prime }(\epsilon ))$ and \(\varepsilon \) $\epsilon$ is stable. In this article, we consider the other extreme case and we prove that, there exist nuclear Fréchet spaces with the properties \((\underline{DN})\) $(\underline{\mathit{DN}})$ and \(\Omega ,\) $\mathrm{\Omega },$ even regular nuclear Köthe spaces, satisfying \(\Delta (E)=\Lambda _{1}(\varepsilon )\) $\mathrm{\Delta }(E)={\mathrm{\Lambda }}_1(\epsilon )$ such that there is no subspace of E which is isomorphic to \(\Lambda _{1}(\varepsilon ).\) ${\mathrm{\Lambda }}_1(\epsilon ).$