No abstract

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. Mathematics Subject Classification (2000). Primary 47B33, Secondary 47B38.

Abstract. We consider composition operators on Hardy spaces of a halfplane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.

Operators on function spaces of form C f D f ı , where is a fixed map are called composition operators with symbol . We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.