Division by inner functions in a class of composition operators on Lipschitz spaces

Abstract

We prove that the class 𝒞 ( H Λ α , H Λ β p ) = { ϕ : C ϕ ( H Λ α ) ⊂ H Λ β p } , where HΛα and H Λ β p are holomorphic Lipschitz spaces, has the f-property for p⩾1, 0<β<1: If ϕ I ∈ 𝒞 ( H Λ α , H Λ β p ) where I is an inner function, then ϕ ∈ 𝒞 ( H Λ α , H Λ β p ) Also we characterize inner functions that belong to 𝒞 ( H Λ α , H Λ β p ) for p>0, 0<α⩽1 and 0<β⩽1. The results are deduced from a new, derivative-free, characterization of 𝒞 ( H Λ α , H Λ β p ) that extends some results of Dyakonov [Acta Math. 178 (1997) 143–167.] and the author [Acta Math. 183 (1999) 141–143; J. Math. Anal. Appl., 326 (2007) 1–11].