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