On some consequences of the isomorphic classification of cartesian products of locally convex spaces

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2016
Kızgut, Ersin
This thesis takes its motivation from the theory of isomorphic classification of Cartesian products of locally convex spaces which was introduced by V. P. Zahariuta in 1973. In the case $X_1 times X_2 cong Y_1 times Y_2$ for locally convex spaces $X_i$ and $Y_i,i=1,2$; it is proved that if $X_1,Y_2$ and $Y_1,X_2$ are in compact relation in operator sense, it is possible to say that the respective factors of the Cartesian products are also isomorphic, up to their some finite dimensional subspaces. Zahariuta s theory has been comprehensively studied for special classes of locally convex spaces, especially for finite and infinite type power series spaces under a weaker operator relation, namely strictly singular. In this work we give several sufficient conditions for such operator relations, and give a complete characterization in a particular case. We also show that a locally convex space property, called the smallness up to a complemented Banach subspace property, whose definition is one of the consequences of isomorphic classification theory, passes to topological tensor products when the first factor is nuclear. Another result is about Fréchet spaces when there exists a factorized unbounded operator between them. We show that such a triple of Fréchet spaces $(X,Z,Y)$ has a common nuclear Köthe subspace if the range space has a property called $(y)$ which was defined by Önal and Terzioğlu in 1990.

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Citation Formats
E. Kızgut, “On some consequences of the isomorphic classification of cartesian products of locally convex spaces,” Ph.D. - Doctoral Program, Middle East Technical University, 2016.