Studies on Function Spaces and unbounded derivations

定义在紧致Hausdorff空间K上的所有连续复值函数空间上的无界导子在K上诱导出某些微分结构。无界导子域可以看作是关于这种结构的所有可微函数的空间，因而是真实的直线上连续可微函数空间的推广.本文研究了无界导子域上的满射线性同构和小界同构.在我们以前的工作中，得到了关于闭导子域上线性等距的结构的几个结果。在本研究中，我们研究了具有sigma范数的闭导子平方域上满射线性等距的结构，并证明了这种等距是由同胚诱导的加权复合算子，在某种意义上，这种同胚可以被看作是K的K的同胚。我们进一步研究了闭导子的ctomain上的小界同构，并证明了在某些假设下，如果存在一个小界同构，则其下的紧Hausdorff空间是同胚的。

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.