The analysis of composition operators on Lp and the Hopf decomposition


Let (X, ∑, μ) be a σ-finite measure space and τ: X → X a measurable transformation. We give an explicit isometric isomorphism between the weighted composition operator induced by a purely dissipative transformation τ and an operator weighted shift. We use this to construct examples of subnormal w.c.o.'s. We show that if a conservative transformation τ is nonsingular, or even if the Radon-Nikodym derivative h = dμ{ring operator} τ-1 dμ is τ-1∑ measurable, then the composition operator induced by τ on L2 is hyponormal precisely when h is τ-invariant. Combined with the invariance of the conservative and dissipative parts from the Hopf decomposition the analysis of Cτ for non-singular τ is simplified. Finally we show that weighted composition operators whose weights are multiplicative coboundaries are isometrically isomorphic to the composition operator induced by the same transformation on the same space with an equivalent measure. This simplifies the analysis of certain examples. © 1991.

Publication Title

Journal of Mathematical Analysis and Applications