Decomposability and the Cyclic Behavior of Parabolic Composition Operators

This paper establishes decomposability for composition operators induced on the Hardy spaces H (1 ≤ p < ∞) by parabolic linear fractional self-maps of the unit disc that are not automorphisms. This result, along with a recent theorem of Miller and Miller [15], shows that no such composition operator is supercyclic. The work here completes part of a previous investigation [1] where the author and Paul Bourdon showed that among linear fractional maps of the disc with no interior fixed point, only the parabolic non-automorphisms induce non-hypercyclic composition operators. Additionally it complements results of Robert Smith [19], who proved decomposability in the case of parabolic automorphisms and it extends recent work of Gallardo and Montes [6] who, using different methods, established non-supercyclicity for non-automorphisms in the case p = 2.