Some thoughts on composition operators on subspaces of the Hardy space

We discuss composition operators on certain subspaces of the Hardy space. The family of subspaces that we deal with are called H alpha,beta 2 which have garnered a lot of attention recently for results related to interpolation. We use them effectively here to study composition operators. Three aspects are discussed. The first is invariance. We examine when H alpha,beta 2 or JH alpha,beta 2 where J is an inner function are left invariant by composition operators. Secondly, we show that for detecting whether a function phi is inner or not, the composition operator with the symbol phi can be used efficiently on certain subspaces. Thirdly, we discover a criterion for detecting invertibility in the footsteps of the classical result of Schwartz.