Weighted composition operators

This thesis contains some results on weighted composition operators on weighted Hardy spaces of the open unit disk D of the complex plane. If ψ is an analytic map on the open unit disk and ϕ an analytic map from the open unit disk into itself, the weighted composition operator Cψ,ϕ is defined by Cψ,ϕ( f)(z) = ψ(z)f(ϕ( z)). These operators come up naturally. Back in 1964 Forelli showed that every isometry on Hp for 1 \u3c p \u3c ∞ and p ≠ 2 is a weighted composition operator [4]. Weighted composition operators have also appeared in recent work of Clifford and Dabkowski [1] and also Shapiro and Smith [8]. Chapter 1 contains definitions and some well known results. The second chapter contains some results on the interaction between ψ and ϕ that is necessary for the boundedness of the operator. The compact operators of this kind are discussed in Chapter 3. A few theorems are given that enable us to construct interesting compact weighted composition operators. The spectrum of a compact operator is also computed. Spectrum of some non-compact operators are discussed in Chapter 4. The last chapter discussus Hilbert-Schmidt operators