Isometric Operators On Analytic Function Spaces

Isometric operator theory in function spaces becomes a hot issue of research in re-cent decades. In order to understand any mathematical structure, one should investigate its group of symmetries. In the class of Banach spaces, such a goal leads naturally to a study of isometries. This paper mainly discusses some properties of isometric operators on some analytic function spaces.1. We study some properties of weighted composition operators on the weighted Dirichlet space of the unit disk. We get the sufficient and necessary condition-s that a weighted composition operator is an isometric operator, and we get the connections among isometric weighted composition operator, isometric composition operator and isometric multiplication operator. Also, we study some properties of isometric composition operator on the Dirichlet space of the poly-disk.2. Firstly, we study the isometric equivalence of composition operator, multiplication operator and weighted composition operator on the space H∞(D), and get their connection when all of the three are isometric equivalent. Then, we study the properties of isometric equivalent composition operators on H∞(D) and HP(D). Finally, we give the properties of isometric equivalent operators on the Bloch space.3. In this chapter, we study the property of isometric composition operator between two different Dirichlet space and the isometric composition operator from weighted Bergman space to Dirichlet space. Also, we study the connections between quasi-isometric composition operator and isometric composition operator on the Dirichlet space.