Approximate local isometries of derivative Hardy spaces

Abstract

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

Mathematics Subject Classification (2020):

• 47B38
• 47B33
• 46B04

Key words:

• Algebraic reflexivity
• topological reflexivity
• isometry group
• isometric reflection