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Operators of Class $$C_0$$ with Spectra in Multiply Connected Regions
Adele Zucchi, Indiana University, Bloomington, IN
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Memoirs of the American Mathematical Society
1997; 52 pp; softcover
Volume: 127
ISBN-10: 0-8218-0626-2
ISBN-13: 978-0-8218-0626-5
List Price: US$41 Individual Members: US$24.60
Institutional Members: US\$32.80
Order Code: MEMO/127/607

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.

• The class $$C_0$$