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Hyponormal powers of composition operators

Authors: Phillip Dibrell and James T. Campbell
Journal: Proc. Amer. Math. Soc. 102 (1988), 914-918
MSC: Primary 47B38; Secondary 47B20
MathSciNet review: 934867
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Abstract: Let $ {T_i},i = 1,2$, be measurable transformations which define bounded composition operators $ {C_{{T_i}}}$ on $ {L^2}$ of a $ \sigma $-finite measure space. Denote their respective Radon-Nikodym derivatives by $ {h_i},i = 1,2$. The main result of this paper is that if $ {h_i} \circ {T_i} \leq {h_j},i,j = 1,2$, then for each of the positive integers $ m,n,p$ the operator $ {[C_{{T_1}}^mC_{{T_2}}^n]^p}$ is hyponormal. As a consequence, we see that the sufficient condition established by Harrington and Whitley for hyponormality of a composition operator is actually sufficient for all powers to be hyponormal.

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