Abstract
We present the current results in the study of weighted composition operators on weighted Banach spaces of an unbounded, locally finite metric space. Specifically, we determine characterizations of bounded and compact weighted composition operators, including the operator and essential norms. In addition, we characterize the weighted composition operators that are injective, are bounded below, have closed range, and have bounded inverse. We characterize the isometries and surjective isometries among the weighted composition operators, as well as those that satisfy the Fredholm condition. Lastly, we provide numerous interesting examples of the richness of these operators acting on the discrete weighted Banach spaces.
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