Approximate local isometries of derivative Hardy spaces

Abstract

For any 1 ≤ p ≤ ∞, let Sp () be the space of holomorphic functions f on such that f′ belongs to the Hardy space Hp (), with the norm ∥f∥∑ = ||f||∞ +||f′|| p . We prove that every approximate local isometry of Sp () is a surjective isometry and that every approximate 2-local isometry of Sp () is a surjective linear isometry. As a consequence, we deduce that the sets of isometric reflections and generalized bi-circular projections on Sp () are also topologically reflexive and 2-topologically reflexive.