Algebraic Reflexivity of Non-Canonical Isometries on Lipschitz Spaces

Abstract

Let Lip([0,1]) be the Banach space of all Lipschitz complex-valued functions f on [0,1], equipped with one of the norms: ∥f∥σ=|f(0)|+∥f′∥L∞ or ∥f∥m=max{|f(0)|,∥f′∥L∞}, where ∥⋅∥L∞ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of Lip([0,1]). Namely, we prove that every approximate local isometry of Lip([0,1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of Lip([0,1]). Additionally, some complete characterizations of such reflections and projections are stated.