Weighted composition operators on discrete weighted Banach spaces

TL;DR

This paper investigates the properties of weighted composition operators on discrete weighted Banach spaces, providing characterizations of boundedness, compactness, isometries, and other operator properties, with numerous illustrative examples.

Contribution

It offers new characterizations and insights into the structure and properties of weighted composition operators on discrete weighted Banach spaces.

Findings

Characterization of bounded and compact weighted composition operators

Identification of isometries and surjective isometries among these operators

Examples demonstrating the diversity of operator behaviors

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.