Journal of Operator Theory

Volume 34, Issue 2, Fall 1995 pp. 197-211.

Construction d’un calcul fonctionnel sous certaines conditions de croissance de la résolvante

Authors: Valérie Martel-Golse
Author institution:École Normale Supérieure - DMI, 45, rue d’Ulm, F-75230 Paris Cedex 05, FRANCE

Summary: Let T be an operator on a Hilbert space H with spectrum $\mathbb T$, the unit circle in $\mathbb C$. Assume the existence of two segments S_1 and S_2 in the complex plane such that $S_i \cap \mathbb T = \{ z_i \}$ and the function $z \to \left\| {(z - T)^{ - 1} } \right\|(1 - \left| z \right|)$ is bounded on $S_1 \cup S_2 \backslash \{z_1 ,z_2 \}$. The main result of the paper is construction of a functional calculus for such operators, defining $\varphi (f)$ for f holomorphic near one of the arcs joining z_1 et z_2 on $\mathbb T$, satisfying f(z_1) = f(z_2) = 0. Various extensions to other types of operators, as well as applications to the invariant subspace problem for operators with connected spectrum are given.

Keywords: Growth of the resolvent, hyperinvariant subspaces, functional calculus, completely non unitary contractions.

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