On Power Series Subspaces of Certain Nuclear Frechet Spaces

TL;DR

This paper investigates the structure of nuclear Fréchet spaces, showing that some have diametral dimensions matching certain power series spaces but do not contain subspaces isomorphic to these spaces, revealing nuanced geometric properties.

Contribution

It demonstrates the existence of nuclear Fréchet spaces with specific diametral dimensions that lack subspaces isomorphic to the corresponding power series spaces, highlighting new structural insights.

Findings

Existence of nuclear Fréchet spaces with prescribed diametral dimensions

Such spaces do not necessarily contain isomorphic subspaces to certain power series spaces

Results extend understanding of subspace structures in nuclear Fréchet 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 $\Lambda_{1}(\varepsilon)$.