Studies on Function Spaces and unbounded derivations

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Unbounded derivations which are defined on the space of all continuous complex valued functions on a compact Hausdorff space K induce certain differential structures on K. Then the domains of unbounded derivations may be regarded as the spaces of all differentiable functions with respect to this structure and, therefore, as one of generalizations of the space of continuously differentiable functions on the real line.In this research, we studied surjective linear isometries and small-bound isomorphisms on such domains. In our previous work, several results on the structure of linear isometries on the domain of closed derivations were obtained. In this research, we studied the structure of surjective linear isometries on the domain of the square of closed derivations equipped with the sigma norm and showed that such isometries are weighted composition operators induced by homeomorphisms )which may be regarded, in a sense, diffeomorphisms of K) of K.We further studied small bound isomorphisms on the ctomain of a closed derivation and showed, under some assumptions, that if there is a small bound isomorphism, then the underlying compact Hausdorff spaces are homeomorphic.