The paper under review investigates algebraic and topological properties of classes of isometries between Lipschitz function spaces (especially algebraic reflexivity and topological reflexivity of subsets of the surjective isometries between \(\text{Lip}(X)\) and \(\text{Lip}(Y)\)).
The authors are interested in a result on linear isometries between Banach spaces of scalar valued Lipschitz functions due to A. Jiménez-Vargas and M. Villegas-Vallecillos [Houston J. Math. 34, No. 4, 1165–1184 (2008; Zbl 1169.46004)], stated as Theorem 1.1 in the paper.
They begin by giving some examples showing that the conditions in Theorem 1.1 are not sufficient for a weighted composition operator to be an isometry. This motivates the sequel of the paper.
Let us mention Theorem 2.1. Let \(X\) and \(Y\) be compact metric spaces.
\((1)\) If there exists an injective real valued function \(f\in \text{Lip}(X)\), then \(\mathcal G_\ast(\text{Lip}(X),\text{Lip}(Y))\) is algebraically reflexive.
\((2)\) If \(Y\) is an \(n\)-dimensional compact and connected manifold without boundary, then \(\mathcal G_\ast(\text{Lip}(X),\text{Lip}(Y))\) is algebraically reflexive.
Here, \(\mathcal G(\text{Lip}(X),\text{Lip}(Y))\) is the set of all surjective linear isometries between \(\text{Lip}(X)\) and \(\text{Lip}(Y)\); and \(\mathcal G_\ast(\text{Lip}(X),\text{Lip}(Y))=\{T\in\mathcal G(\text{Lip}(X),\text{Lip}(Y)) : T(1_X)\) is a nonvanishing contraction\(\}\).
On the other hand, we also mention Proposition 3.1, asserting that, under some specific compactness hypothesis on \(X\) and \(Y\), an operator \(T\in\mathcal L(\text{Lip}(X),\text{Lip}(Y))\) is surjective provided that \(T\) is a topologically surjective isometry.
As a corollary (see Corollary 3.1), under the same assumptions on \(X\) and \(Y\), the space \(\mathcal G_1(\text{Lip}(X),\text{Lip}(Y))\) of \(T\in\mathcal G_\ast(\text{Lip}(X),\text{Lip}(Y))\) preserving unities, is topologically reflexive.