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

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.


On the isomorphic classification of the cartesian products of köthe spaces
Taştüner, Emre; Yurdakul, Murat Hayrettin; Department of Mathematics (2019)
In 1973, V. P. Zahariuta formed a method to classify the Cartesian products of locally convex spaces by using the theory of Fredholm operators. In this thesis, we gave modifications done in the method of Zahariuta. Then by using them, we studied the isomorphic classifications of Cartesian products of l^p and l^q type Köthe sequence spaces.
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Uyanık, Elif; Yurdakul, Murat Hayrettin; Department of Mathematics (2017)
In this thesis we study on bounded and unbounded operators and obtain some results by considering $ell$-K"{o}the spaces. As a beginning, we introduce some necessary and sufficient conditions for a Cauchy Product map on a smooth sequence space to be continuous and linear and we consider its transpose. We use the modified version of Zahariuta's method to obtain analogous results for isomorphic classification of Cartesian products of K"{o}the spaces. We also investigate the SCBS property and show that all sepa...
Strictly singular operators and isomorphisms of Cartesian products of power series spaces
Djakov, PB; Onal, S; Terzioglu, T; Yurdakul, Murat Hayrettin (1998-01-02)
V. P. Zahariuta, in 1973, used the theory of Fredholm operators to develop a method to classify Cartesian products of locally convex spaces. In this work we modify his method to study the isomorphic classification of Cartesian products of the kind E-0(p)(a) x E-infinity(q) (b) where 1 less than or equal to p, q < infinity, p not equal q, a = (a(n))(n=1)(infinity) and b = (b(n))(n=1)(infinity) are sequences of positive numbers and E-0(p)(a), E(infinity)q(b) are respectively l(p)-finite and l(q)-infinite type...
On a problem of Osserman in Lorentzian geometry
GarciaRio, E; Kupeli, DN; VazquezAbal, ME (Elsevier BV, 1997-03-01)
A problem of Osserman on the constancy of the eigenvalues of the Jacobi operator is studied in Lorentzian geometry. Attention is paid to the different cases of timelike, spacelike and null Osserman condition. One also shows a relation between the null Osserman condition and a previous one on infinitesimal null isotropy.
On the Krall-type polynomials on q-quadratic lattices
Alvarez-Nodarse, R.; Adiguzel, R. Sevinik (Elsevier BV, 2011-08-01)
In this paper, we study the Krall-type polynomials on non-uniform lattices. For these polynomials the second order linear difference equation, q-basic series representation and three-term recurrence relations are obtained. In particular, the q-Racah-Krall polynomials obtained via the addition of two mass points to the weight function of the non-standard q-Racah polynomials at the ends of the interval of orthogonality are considered in detail. Some important limit cases are also discussed. (C) 2011 Royal Net...
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.