On the Dynamics of Lipschitz Operators

Abstract

We consider weighted composition operators, that is operators of the type g → w • g • f , acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality.

Injectivity of Lipschitz Operators

“…Proof. Let M = {0, 1} ∪ {x n : n ∈ N} ∪ {y n : n ∈ N}, where x 2 n−1 +k = 2 n−1 + k + 1 and y 2 n−1 +k = x 2 n−1 +k + 1 4 n , k = 0, . .…”