Surjective isometries on an algebra of analytic functions with $C^n$-boundary values (Research on preserver problems on Banach algebras and related topics)

Abstract

Let 𝔻, 𝔻⁻ and 𝕋 be the open unit disk, closed unit disk and unit circle in ℂ. Let $A^n$(𝔻⁻) denote the algebra of all continuous functions f on 𝔻⁻ which are analytic in 𝔻 and whose restrictions f|𝕋 to T are of class $C^n$. For each f ∈ $A^n$(𝔻⁻), the k-th derivative of f|𝕋 as a function on 𝕋 is denoted by D^k(f). We characterize surjective, not necessarily linear, isometries on $A^n$(𝔻⁻) with respect to the norm ∥f∥𝔻⁻ + Σ[n][k=1]∥$D^k$(f)∥𝕋/k!, where ∥ · ∥𝔻⁻ and ∥ · ∥𝕋 are the supremum norms on 𝔻⁻ and 𝕋, respectively.