Normal weighted composition operators on the Hardy space H2(U)

Let phi be an analytic function on the open unit disc U such that phi(U) subset of U, and let psi be an analytic function on U such that the weighted composition operator W-psi,W-phi defined by W-psi,W-phi f = psi f circle phi is bounded on the Hardy space H-2(U). We characterize those weighted composition operators on H-2(U) that are unitary. showing that in contrast to the unweighted case (psi equivalent to 1), every automorphism of U induces a unitary weighted composition operator. A conjugation argument, using these unitary operators, allows us to describe all normal weighted composition operators on H-2(U) for which the inducing map phi fixes a point in U. This description shows both psi and phi must be linear fractional in order for W-psi,W-phi to be normal (assuming phi fixes a point in U). In general, we show that if W-psi,W-phi is normal on H-2 (U) and psi not equivalent to 0, then phi must be either univalent on U or constant. Descriptions of spectra are provided for the operator W-psi,W-phi : H-2(U) -> H-2(U) when it is unitary or when it is normal and phi fixes a point in U. (C) 2010 Elsevier Inc. All rights reserved.