Composition Operators

Abstract

The object of this note is to report on some of the properties of a class of operators induced by inner functions. If m is normalized Lebesgue measure on the unit circle X in the complex plane and Cϕ is an inner function (a complex function on X of unit modulus almost everywhere whose Poisson integral is a non-constant holomorphic function in the open unit disk), then an operator Cϕ on L2(m) is defined by