Subnormality of Weighted Shifts Associated to Composition Operators

摘要

Let phi be a linear fractional transformation mapping the unit disk D into itself and fixing 1, and let C phi be the usual composition operator induced by phi on the Hardy space H2(D). For a point z of D, the subspace generated by the reproducing kernels k phi n(z) (n=0,1,...) arising from iterates of z under phi is invariant for the adjoint C phi. On this subspace, we show that C phi is similar to a weighted shift, and the subnormality and hyponormality of the shift or their lack (which, surprisingly, coincide) are determined by z and the location of the other fixed point of phi in pleasing geometrical ways. These results are applied to semigroups of composition operators arising from linear fractional transformations.