Studies of the Structure of Operators on Function Spaces

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The theory of function spaces plays an basic role in many branches of pure mathematics, for example, function analysis, differential equations, Fourier analysis, etc. and has become a useful tool in applied mathematics.In this research, we studied the space of diffrentiable functions from the point of view of unbounded derivations.The domain of a unbounded derivation in the space of continuous functions on a compact Hausdoruff space may be regarded as one of generalizations of the space of continuously differentiable functions. We investigated the structure of two important operators (that is, surjective linear isometries and small-bound isomorphisms) on such domain.At first, we decided extreme points of the unit sphere of the conjugate space of the domain equipped with the Cambern norm and used it to prove that surjective linear isometries of the domain are weighted composition operators induced by homeomorphisms of the underlying compact Hausdorff spaces. When the underlying topological spaces satisfy the first countability axiom, this result is extended to the second order derivations.We further studied small-bound isomorphisms on the domain and showed that if there is a small bound-isomorphism, then the underlying topological spaces are homeomorphic.As by-product, we obtained Korovkin type approximation theorems on the space of continuously differentiable functions on the unit interval of the real line.