Weighted composition operators on Hardy–Smirnov spaces

Abstract

Abstract Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.