Weighted composition operators between different Hardy spaces. (English) Zbl 1042.47017

Contreras, Manuel D.; Hernández-Díaz, Alfredo G.

Integral Equations Oper. Theory 46, No. 2, 165-188 (2003).

Let \(H(D)\) be the space of analytic functions \(f:D\to{\mathbb C}\) where \(D\) is the usual open unit disk in \(\mathbb C\). Let \(\varphi\in H(D)\) with \(\varphi(D)\i D\) and \(\psi\in H(D)\) be given. The weighted composition operator \(W_{\varphi,\psi}:H(D)\to H(D)\) is defined by \(f\mapsto\psi\cdot(f\circ\varphi)\). In this paper, such operators are considered on the classical Hardy spaces \(H^p\). Specifically, the question is investigated of when, given \(1\leq p,q<\infty\), \(W_{\varphi,\psi}\) maps \(H^p\) boundedly into \(H^q\). Characterizations of compactness, weak compactness, and complete continuity of such operators are obtained which generalize those known for the usual composition operators. The authors apply their results to investigate composition operators between Hardy spaces on the upper half-plane in \(\mathbb C\).

Reviewer: Hans Jarchow (Zürich)

MSC:

47B33	Linear composition operators
46E15	Banach spaces of continuous, differentiable or analytic functions
30D55	\(H^p\)-classes (MSC2000)

Keywords:

weighted composition operator; classical Hardy spaces; compactness; weak compactness; complete continuity