OPERATORS OF CLASS C-0 WITH SPECTRA IN MULTIPLY CONNECTED REGIONS

mLet Omega be a bounded finitely connected region ia the complex plane , whose boundary Gamma consists of disjoint, analytic, simple closed c urves. We consider linear bounded operators on a Hilbert space H havin g <(Omega)over bar> as spectral set, and no normal summand with spectr um in Gamma. For each operator satisfying these properties, we define a weak-continuous functional calculus representation Phi: H-infinity( Omega) --> L(H), where H-infinity(Omega) is the Banach algebra of boun ded analytic functions on: SZ, An operator is said to be of class Co i f the associated functional calculus has a non-trivial kernel. In this paper we study operators of class C-0, for which we provide a complet e classification into quasisimilarity classes, analogous to the; case of the unit disk.