Some remarks on substitution and composition operators

Abstract

In this paper we study the linear substitution operator Sϕ(f) := f ◦ ϕ generated by some function ϕ : [0, 1] → [0, 1], as well as the nonlinear composition operator Cg(f) := g ◦ f generated by some function g : R → R. We will show that these operators have a very different (and sometimes quite surprising) behavior in the space of con-tinuous functions, Lipschitz functions, functions of bounded variation, and Baire class one functions. A main emphasis is put on examples and counterexamples which illustrate this behavior.