Complex Analysis and Operator Theory
Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces.
Matache, Valentin, "Weighted Composition Operators on H² and Applications" (2008). Mathematics Faculty Publications. 40.