Discrete Measure Spaces

In this chapter, we adapt our general results to the context of discrete weighted composition operators, i.e., weighted composition operators over discrete measure spaces. Section 6.1 has an introductory character. Section 6.2 characterizes the hyponormality, cohyponormality and normality of discrete weighted composition operators (see Theorems 83, 84 and 87). Section 6.3 provides two criteria for the subnormality of discrete weighted composition operators, the second of which generalizes the discrete version of one of Lambert’s characterizations of bounded subnormal composition operators (see Theorems 89 and 90). The interplay between the theory of moments, the geometry of graphs induced by symbols and the injectivity problem is discussed in Section 6.4 (see Theorem 93 and Problems 96 and 100). Section 6.5 contains a variety of examples illustrating our considerations.