Approximate local isometries of derivative Hardy spaces

Abstract

For any p in [1,\infty], let S^p(D) be the space of holomorphic functions f on D such that f' belongs to the Hardy space H^p(D), with the sum norm. We prove that every approximate local isometry of S^p(D) is a surjective isometry and that every approximate 2-local isometry of S^p(D) is a surjective linear isometry. As a consequence, we deduce that the sets of isometric reflections and generalized bi-circular projections on S^p(D) are also topologically reflexive and 2-topologically reflexive.