Weakly compact weighted composition operators on spaces of Lipschitz functions

On the Dynamics of Lipschitz Operators

“…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]).…”

“…Another important ingredient is Theorem 2.14 which allows us to prove that weak compactness is actually equivalent to (norm) compactness for the class of operators we consider in this article. The latter result in an improvement of [2,Theorem B] where w = 1, and [19,Theorem 2.1] where M is a compact purely 1-unrectifiable metric space. The two key elements mentioned above permit us to deduce a new and general characterization of (weak) compactness for weighted Lipschitz/composition operators.…”