Weighted composition operators between Lipschitz spaces on pointed metric spaces

Abstract

In this paper, we study weighted composition operators between Banach spaces of scalar-valued Lipschitz functions on pointed metric spaces, not necessarily compact. We give necessary and sufficient conditions for the injectivity and the surjectivity of these operators. We also obtain sufficient and necessary conditions for a weighted composition operator between these spaces to be compact.

On the Dynamics of Lipschitz Operators

“…Remark 3.8. When M is bounded, w needs not be Lipschitz nor bounded for wC f to be bounded from Lip 0 (N ) to Lip 0 (M ); an example can be found in [13] (see Example 2 therein). But, in this case, Lip 0 (M ) can be seen as 1-codimensional subspace of Lip(M ) (isomorphically speaking).…”

“…(2) Assuming moreover that w is a Lipschitz map, one can deduce the next simpler statement (which corresponds to [13,Theorem 4.3]): w f and wC f are compact if and only…”

“…In the same paper [9], under the hypothesis that f (M ) it totally bounded in N , the authors provide a necessary and sufficient condition for such operators to be compact (see also [13] for a similar result with the additional assumption w ∈ Lip(M )). Finally, injectivity and surjectivity of such operators are characterized in [13] (see also [18] for compact M ), while weak compactness is considered in [19]. In the latter paper, M is assumed to be a compact metric space such that the little Lipschitz space has the uniform separation property (these metric spaces are characterized in [4, Theorem A]).…”

“…A complete description of bounded weighted composition operators from Lip 0 (N ) to Lip 0 (M ) has recently been given in [9] (see also [13] in the case when M is bounded). In the same paper [9], under the hypothesis that f (M ) it totally bounded in N , the authors provide a necessary and sufficient condition for such operators to be compact (see also [13] for a similar result with the additional assumption w ∈ Lip(M )). Finally, injectivity and surjectivity of such operators are characterized in [13] (see also [18] for compact M ), while weak compactness is considered in [19].…”

“…Finally, in subsections 4.1 and 4.2, we focus on compactness but this time with stronger assumptions on either M , f : M → N or w : M → K. The outcome is that we are able to provide nicer statements than Theorem A.1. Importantly, we retrieve the main results from [9] and [13].…”