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Compound Invariants and Mixed F-, DF-Power Spaces
Published online by Cambridge University Press: 20 November 2018
Abstract
The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed $F-$, $\text{DF}$-power series spaces, i.e. the spaces of the following kind
where ${{a}_{i}}(p,\,q)\,=\,\exp \left( \left( p\,-\,{{\lambda }_{i}}q \right){{a}_{i}} \right),\,p,\,q\,\in \,\mathbb{N},\,\text{and}\,\lambda \,\text{=}\,{{\left( {{\lambda }_{i}} \right)}_{i\in \mathbb{N}}},\,a=\,{{({{a}_{i}})}_{i\in \mathbb{N}}}$ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of $F-$ and $\text{DF}$-types, respectively. The m-rectangle characteristic $\mu _{m}^{\lambda ,a}\left( \delta ,\,\varepsilon ;\,\tau ,\,t \right),\,m\,\in \,\mathbb{N}$ of the space $G\left( \text{ }\!\!\lambda\!\!\text{ ,}\,a \right)$ is defined as the number of members of the sequence ${{({{\lambda }_{i}},{{a}_{i}})}_{i\in \mathbb{N}}}$ which are contained in the union of m rectangles ${{P}_{k}}\,=\,\left( {{\delta }_{k}},\,{{\varepsilon }_{k}} \right]\,\times \,\left( {{\tau }_{k}},\,{{t}_{k}} \right]$ , $k\,=\,1,2,\ldots ,m$. 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, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).
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- Copyright © Canadian Mathematical Society 1998
Footnotes
Permanent address: Department of Mechanics and Mathematics Rostov State University Rostov-on-Don Russia
The first author was supported by Tübİtak-Nato Fellowship Program.
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