Approximate local isometries of derivative hardy spaces

Abstract

For any 1 ≤ p ≤ ∞, let S^p (ⅅ) be the space of holomorphic functions ƒ on ⅅ such that ƒ′ belongs to the Hardy space H^p(ⅅ), with the norm ∥ƒ∥_∑ = ∥f∥_∞+∥ƒ′∥_p. 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 re ections and generalized bi-circular projections on S^p(ⅅ) are also topologically re exive and 2-topologically reflexive.