Weighted Composition Operators on Weighted Hardy Spaces

For functions mu : N-0 -> C and theta : N-0 -> N-0, the weighted composition operator is the operator Gamma(mu,theta) : H-beta(p) -> C-N0 defined by Gamma(mu,theta)(x) = Sigma(k)(m=0) mu( k - m) x(theta(m)))(k=0)(infinity) for all x = (x(k)) is an element of H-beta(p). This paper deals with the properties of weighted composition operators on weighted Hardy spaces H-beta(p) for 1 <= p <= infinity. The necessary and sufficient conditions are investigated for a weighted composition operator to be compact. A compact subset of non-negative real numbers containing the essential norm of Gamma(mu,theta) on H-beta(p) is also computed. The condition under which weighted composition operators commute is also explored.