Abstract
We prove here some mapping theorems for the operators of class Cp (p > 0) on Hilbert spaces defined by B. Sz-Nagy and C. Foias in [12]. More precisely, we answer the following question: What can be said about membership of f(T) in the classes Cp (H) when T belongs to a given one of them and f is in the disc algebra? As a corollary, we recover in this way the famous power inequality. We also introduce a Harnack ordering relation between such operators, we define the corresponding Harnack parts in the class Cp, and we give some results related to uniformly stable operators. These parts involve some operatorial Harnack inequalities as well as von Neumann inequalities, which generalize the Harnack inequalities for contractions given by C. Foias [8], K. Fan [5] and I. Suciu [17]. Finally, we define the Harnack and hyperbolic metrics, and we prove that these are complete on the set of all uniformly stable operators, respectively on the Harnack parts of such operators.
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