# Operators of Class \(C_0\) with Spectra in Multiply Connected Regions

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*Adele Zucchi*

Let \(\Omega\) be a bounded finitely connected region in the complex plane, whose boundary \(\Gamma\) consists of disjoint, analytic, simple closed curves. The author considers linear bounded operators on a Hilbert space \(H\) having \(\overline \Omega\) as spectral set, and no normal summand with spectrum in \(\gamma\). For each operator satisfying these properties, the author defines a weak\(^*\)-continuous functional calculus representation on the Banach algebra of bounded analytic functions on \(\Omega\). An operator is said to be of class \(C_0\) if the associated functional calculus has a non-trivial kernel. In this work, the author studies operators of class \(C_0\), providing a complete classification into quasisimilarity classes, which is analogous to the case of the unit disk.