Title: Mapping theorems and Harnack ordering for \rho-contractions

Authors: Gilles Cassier and Nicolae Suciu

Issue: Volume 55 (2006), Issue 2, 483-524

Abstract: We prove here some mapping theorems for the operators of class $C_{\rho}$ ($\rho > 0$) on Hilbert spaces defined by B. Sz-Nagy and C. Foias in \textit{Analyse Harmonique des Op\'erateurs de l'espace de Hilbert} (Paris: Masson, 1967). More precisely, we answer the following question: What can be said about membership of $f(T)$ in the classes $C_{\rho}(H)$ 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 $C_{\rho}$, 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 (\textit{On Harnack parts of contractions}, Rev. Roumaine Math. Pures Appl. \textbf{19} (1974), 315--318), K. Fan (\textit{Analytic functions of a proper contraction}, Math. Z. \textbf{160} (1978), 275--290), and I. Suciu (\textit{Harnack inequalities for a functional calculus}, in: "Hilbert Space Operators and Operator Algebras (Proc. Internat. Conf., Tihany, 1970)" (Amsterdam: North-Holland, 1972), pp. 499-511). 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.