Reducibility of Composition Operators on L2

Given a sigma finite measure space (X, A, m) and a measurable transformation T: X → X, the reducibility of the composition operator C: ƒ → ƒ ∘ T on L2 is examined. It is shown that if C is reducible then either there is a subset A of X such that L2(A, A, m) reduces C, or there is a sigma subalgebra B of A such that L2(X, B, m) reduces C. Several examples are then presented illustrating various levels of reducibility for composition operators.