
Compound invariants and mixed $F$, $\DF$power spaces
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).
Categories:46A04, 46A45, 46M05 