CONNECTED COMPONENTS IN THE SPACE OF COMPOSITION OPERATORS ON H FUNCTIONS OF MANY VARIABLES

Let E be a complex Banach space with open unit ball BE. The structure of the space of composition operators on the Banach algebra H (BE), of bounded analytic functions on BE with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly inside BE form a path connected component. When E is a Hilbert space or a Co(X)- space, the path connected components are shown to be the open balls of radius 2.