Normal and invertible composition operators

Abstract

Let N denote the set of natural numbers and let φ be a mapping from N into itself. Then the composition transformmation Cφ, on the weighted l2 space with weights a2n, where n ∈ N and 0 < a < 1 is defined by Cφf = f ∘ φ. If Cφ is a bounded operator, then it is called a composition operator. The adjoint of the composition operator Cφ, is computed, and it is used to characterise normal, unitary, isometric, and co-isometric composition operators. Not every invertible φ induces an invertible composition operator, as is shown by examples. At the end of this note all invertible composition operators are characterised.