Compact composition operators

Abstract

Abstract Let (Хζ,λ) be a σ-finite measure space, and let ϕ be a non-singular measurable transformation from X into itself. Then a composition transformation C ϕ on L 2 (λ) is defined by C ϕ f = f ∘ ϕ. If C ϕ is a bounded operator, then it is called a composition operator. The space L 2 (λ) is said to admit compact composition operators if there exists a ϕ such that C ϕ is compact. This note is a report on the spaces which admit or which do not admit compact composition operators.