Decomposability and the cyclic behavior
of parabolic composition operators

Joel H. Shapiro

 

In: Recent Progress in Functional Analysis, Proceedings of the International Functional Analysis Meeting on the Occasion of the 70th Birthday of Professor Manuel Valdivia, Valencia, Spain, July 3-7, 2000, K.D. Bierstedt, J. Bonet, M. Maestre, J. Schmets (ed.), North-Holland Math. Studies, 2001.

 

Abstract: In this paper I study composition operators induced by linear fractional selfmaps of the unit disc that are parabolic, but not onto. I show that such maps induce composition operators on the Hardy space H^p (1 <= p < infinity) that are decomposable. This result, the spectral properties of the operators being studied, and a recent result of Len and Vivien Miller, shows that such composition operators are not supercyclic. This work generalizes previous work of Gallardo and Montes, who proved the non-supercyclicity result, by different methods, for p=2, and it complements work of Robert Smith who proved decomposability for composition operators induced by parabolic automorphisms ("onto parabolics"). The paper features a significant expository component that makes it mostly self-contained.

 

Download .pdf file (180K):