Isometries Of Composition Operators On Some Function Spaces

Operator theory in function spaces becomes a hot issue of research in recentdecades. As one of the important parts of operator theory, the research about isome-try of composition operator has focused on the attentions of the researchers. So, moreand more the mathematics researchers start to study it. This paper mainly discussesthe isometries on products of composition and integral operator and products of com-position and differentiation operator and Generalized composition operator on somefunction spaces.First, we discuss the isometries on products of composition and integral operatorsC(?)Jg, JgC(?) from the space of weighted bounded analytic function to Bloch type spaceon the unit disk, and give necessary and suffcient conditions, which C(?)Jg, JgC(?) areisometric operators. Additional, we study the isometries on products of compositionand integral operators C(?)Ig, IgC(?) on Bloch type space, and we give necessary andsuffcient conditions, which IgC(?),C(?)Ig are isometric operators.Next, we discuss the isometries on products of composition and differentiationoperators DC(?),C(?)D from Bloch type space to the space of weighted bounded ana-lytic function on the unit disk, and we give necessary and suffcient conditions whichDC(?),C(?)D are isometric operators.Finally, we discuss the isometry on Generalized composition operator C(?)g on Blochtype space on the unit disk, and we give a necessary and suffcient condition which C(?)g isa isometric operator. Additional, we define a new Generalized operator Cgff and discussthe isometry of it on Bloch type space, we give a necessary and suffcient conditionwhich Cgff is a isometric operator.