On bounded and unbounded operators

Uyanık, Elif
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 separable Fr{'e}chet-Hilbert spaces have this property. By the help of this result, we obtain that the bounded perturbation of an automorphism on a separable Fr{'e}chet-Hilbert space still takes place up to a complemented Hilbert subspace. We also show that the strong dual of a Fr{'e}chet-Hilbert space has the SCBS property. After that, we consider $ell$-K"{o}the spaces and obtain necessary and sufficient condition for every continuous linear operator from a Fr{'e}chet space to $ell$-K"{o}the space to be bounded. In addition, we obtain a sufficient condition when each continuous linear operator from a Fr{'e}chet space $X$ to $ell$-K"{o}the space $lambda^{ell_3}(C)$ that factors over the projective tensor product of $ell$-K"{o}the spaces $lambda^{ell_1}(A) hat{otimes}_{pi} lambda^{ell_2}(B)$ is bounded when $lambda^{ell_1}(A)$ and $lambda^{ell_2}(B)$ are nuclear. We show that if there exists a continuous linear unbounded operator between $ell$-K"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K"{o}the matrices when every continuous linear operator between $ell$-K"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between $ell$-K"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace. 


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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 ...
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Citation Formats
E. Uyanık, “On bounded and unbounded operators,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.