Several Types Of Function Space, Get The Child The Nature Of Discussion

The composition operator is defined on the functional space. It has a close relation with functional theory. In fact, a lot of problems in the functional theory are corresponding to the problems in the composition operator.A lot of investigations focus on the two kinds of functional spaces. One functional space is Hardy space, which is defined on the complex plane, the other is Hardy space but is defined on the ball in the C n.And there are many results. In the C n, it is more difficult to obtain these results. And there are a lot of important results in the composition operator, such as the composition operators which are defined on Hardy space and Bloch space. The generalization of composition operator, one is the weighted composition operator, the other is the generalization of the functional space. There are new techniques and problems. In the investigations, the boundedness, compactness of the composition operator C? is very important.In the article, we investigate the boundedness, compactness of the composition operator C? in detail. By the investigation, we know the characteristic about boundedness, compactness of the composition operator. The main results are as follows:1. We use the tool of Carleson measure. By the Carleson measure, we investigate the compactness of the composition operator C? which is mapped into E ( p , q ) space. At first, we prove a theorem which is a characteristic about the compactness of C? .We discuss the induced Borel measure, we use the property of compact Carleson measure to characterise the compactness about the composition operator C? .2. In Hardy space, the reversibility, Fredholmness of composition operator C? is equivalent to the symbol ? is a Mobius function on the disk. We investigate the problem on theβp space, we prove the reversibility, Fredholmness of composition operator C? is equivalent to the symbol ? is a Mobius function on the disk.3. At last, we investigate the boundedness and compactness of operator C? ,which is defined on a little weighted Bloch space. We give a characteristic about the bounded set and relatively compact set. Then we give the characteristic about the boundedness, compactness of the composition operator.