Compact and Hilbert-Schmidt weighted composition operators on the Hardy space

The classical Hardy space H2(D ) is the Hilbert space of analytic functions on the open unit disk D in the complex plane such that the Taylor series coefficients of each analytic function f in H2( D) are square-summable. If ϕ is an analytic self-map of the open unit disk D then the composition operator Cϕ on the Hardy space is defined by (Cϕf)(z) = f(ϕ(z)) where f is in H2(D) and z is in D. If psi ∈ H 2(D) then the weighted composition operator Wpsi,ϕ on the Hardy space H 2(D) is given by (Wpsi,ϕf)( z) = psi(z)f(ϕ( z)). In this case ϕ is called the inducing map and psi is called the weight function. The goal in the study of weighted composition operators is to relate the operator-theoretic properties of W psi,ϕ to the function-theoretic properties of the inducing map ϕ and the weight function psi.; An operator-theoretic property of Wpsi,ϕ that is of interest to us is the compactness of Wpsi,ϕ . The operator Wpsi,ϕ is said to be compact if it maps bounded sets in H2(D ) into relatively compact sets. Gunatillake in his dissertation gave a sufficient condition for Wpsi,ϕ to be compact. In this dissertation we will present some necessary conditions, some sufficient conditions and some necessary and sufficient conditions with restrictions on psi for which Wpsi,ϕ is compact. An important subclass of compact operators on a Hilbert space is the class of Hilbert-Schmidt operators. We will characterize the Hilbert-Schmidt property of Wpsi,ϕ for some special classes of the weight function psi and inducing map psi.