The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed $F$-, $\DF$-power series spaces, {\it i.e.} the spaces of the following kind $$ G(\la,a)=\lim_{p \to \infty} \proj \biggl(\lim_{q \to \infty}\ind \Bigl(\ell_1\bigl(a_i (p,q)\bigr)\Bigr)\biggr), $$ where $a_i (p,q)=\exp\bigl((p-\la_i q)a_i\bigr)$, $p,q \in \N$, and $\la =( \la_i)_{i \in \N}$, $a=(a_i)_{i \in \N}$ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of $F$- and $\DF$-types, respectively. The $m$-rectangle characteristic $\mu_m^{\lambda,a}(\delta,\varepsilon; \tau,t)$, $m \in \N$ of the space $G(\la,a)$ is defined as the number of members of the sequence $(\la_i, a_i)_{i \in \N}$ which are contained in the union of $m$ rectangles $P_k = (\delta_k, \varepsilon_k] \times (\tau_k, t_k]$, $k = 1,2, \ldots, m$. It is shown that each $m$-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pe{\l}czynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).

Compound invariants and mixed $F$-, $\DF$-power spaces

Publisher Summary This chapter discusses the applications of a decomposition method. Let E be a specific locally convex space. By assuming (1) F is isomorphic to a complemented subspace of E and (2) E is isomorphic to a complemented subspace of F , it can be concluded that F is in fact isomorphic to E ; thus, there is a decomposition method for E . Because E is a concrete space, the problem of imbedding F as a complemented subspace into E can be expected to be easier than imbedding E into F as a complemented subspace. The chapter discusses the problem of imbedding nuclear power series spaces of infinite type.

Some applications of a decomposition method

The diametral dimension, $\Delta(E)$, and the approximate diametral dimension, $\delta (E)$, of a nuclear Frechet space $E$ which satisfies $\underline{DN}$ and $\Omega$, are related to corresponding invariant of power series spaces $\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}\left(\varepsilon\right)$ for some exponent sequence $\varepsilon$. In this article, we examine a question of whether $\delta (E)$ must coincide with that of a power series space if $\Delta(E)$ does the same, and vice versa. In this regard, we first show that this question has an affirmative answer in the infinite type case by showing that $\Delta (E)=\Delta\left(\Lambda_{\infty} (\varepsilon)\right)$ if and only if $\delta (E)= \delta (\Lambda_{\infty}(\varepsilon))$. Then we consider the question in the finite type case and, among other things, we prove that $\delta (E)=\delta\left(\Lambda_{1} (\varepsilon)\right)$ if and only if $\Delta (E)= \Delta (\Lambda_{1}(\varepsilon))$ and $E$ has a prominent bounded subset.

Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fréchet Spaces

The diametral dimension, $\Delta(E)$, and the approximate diametral dimension, $\delta (E)$, of a nuclear Frechet space $E$ which satisfies $\underline{DN}$ and $\Omega$, is related to power series spaces $\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}\left(\varepsilon\right)$ for some exponent sequence $\varepsilon$. In this article, we examine a question of whether $\delta (E)$ must coincide with that of a power series space if $\Delta(E)$ does the same, and vice versa. In this regard, we first show that this question has an affirmative answer in an infinite type case by proving the fact that $\Delta (E)=\Delta\left(\Lambda_{\infty} (\varepsilon)\right)$ if and only if $\delta (E)= \delta (\Lambda_{\infty}(\varepsilon))$. Then we consider the question in the finite type case and, among other things, we prove that $\delta (E)=\delta\left(\Lambda_{1} (\varepsilon)\right)$ if and only if $\Delta (E)= \Delta (\Lambda_{1}(\varepsilon))$ and $E$ has a prominent bounded subset.

Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\'echet Spaces

The purpose of this article is to give an exposition of some recent results related to imbedding smooth sequence spaces into nuclear Frechet spaces. Although some new results are stated and proved in the present article, essentially they are modifications of the results of Aytuna, Krone and the author which are contained in [6] and [7].

Some Invariants of Frechet Spaces and Imbeddings of Smooth Sequence Spaces

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Some Invariants of Frechet Spaces and Imbeddings of Smooth Sequence Spaces