COMPOSITION OPERATORS ON WEIGHTED HARDY SPACES

Suppose phi is an analytic self-map of open unit disk D and psi is an analytic function on D. Then a weighted composition operator induced by psi with weight is given by (W(psi phi)f)(z) = psi(z) f ((phi(z)), for z is an element of D and f analytic on D. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators W-psi,W-phi. In terms of fixed points in the closed unit disk D, conditions under which W-psi,W-phi is compact are given. Necessary conditions for the compactness of C-phi, are given in terms of the angular derivative phi'(zeta) where zeta is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten p-class S-p(H-s(beta(1)), H-q(beta(2))), where the inducing map has supremum norm strictly smaller than 1.