Normal and quasinormal composition operators

Abstract

A bounded linear operator C T {C_T} on L 2 ( X , Σ , m ) {L^2}(X,\Sigma ,m) is a composition operator if it is induced by a point mapping T : X → X T:X \to X via C T f = f ∘ T {C_T}f = f \circ T . Normal and quasinormal composition operators on a finite measure space are characterized: C T {C_T} is normal iff T is measure preserving and T − 1 ( Σ ) {T^{ - 1}}(\Sigma ) is (essentially) all of Σ ; C T \Sigma ;{C_T} is quasinormal iff T is measure preserving.