| Popis: |
In this chapter, we investigate the interplay between selected properties of a weighted composition operator Cϕ,w and the corresponding composition operator C ϕ . In Sect. 7.1, we discuss the questions of when the product M w C ϕ is closed and when it coincides with Cϕ,w (see Theorems 110 and 112). The relationships between the Radon-Nikodym derivatives h ϕ and hϕ,w are described in Sect. 7.2 (see Propositions 116, 119, and 121). In Sect. 7.3, using a result due to Berg and Duran, we give conditions enabling us to deduce the subnormality of Cϕ,w from that of C ϕ (see Theorem 126). The converse possibility is discussed in Theorem 130. In Sect. 7.4, we provide a criterion for a bounded weighted composition operator with matrix symbol to be subnormal (see Theorem 131). Section 7.5 contains numerous examples illustrating our considerations. |
