Surjective isometries on a Banach space of analytic functions on the open unit disc

Abstract

Let $\mathcal{S}_A$ be the complex linear space of all analytic functions on the open unit disc $\mathbb D$, whose derivative can be extended to the closed unit disc $\bar{\mathbb D}$. We give the characterization of surjective, not necessarily linear, isometries on $\mathcal{S}_A$ with respect to the norm $\| f \| _{\sigma} = |f(0)| + \sup \{|f'(z)| : z \in \mathbb D \}$ for $f \in \mathcal{S}_A$.