Compactness and related properties of weighted composition operators on weighted BMOA spaces

It is shown that a large class of properties coincide for weighted composition operators on a large class of weighted VMOA spaces, including the ones with logarithmic weights and the ones with standard weights $(1-\mathrm{\mid }z\mathrm{\mid }{)}^{-c},0\le c<1\mathrm{/}2$. Some of these properties are compactness, weak compactness, complete continuity and strict singularity. A function-theoretic characterization for these properties is also given. Similar results are also proved for many weighted composition operators on similarly weighted BMOA spaces. The main results extend the theorems given in [Proc. Amer. Math. Soc. 151 (2023), 1195–1207], and new test functions that are suitable for the weighted setting are developed.