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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Essential spectra of operators in the class $ \mathcal{B}_n(\Omega)$

Author: Karim Seddighi
Journal: Proc. Amer. Math. Soc. 87 (1983), 453-458
MSC: Primary 47A53; Secondary 47B38
MathSciNet review: 684638
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Abstract: For a connected open subset $ \Omega $ of the plane and $ n$ a positive integer, let $ {\mathcal{B}_n}(\Omega )$ be the space introduced by Cowen and Douglas in their paper Complex Geometry and Operator Theory. Our paper deals with characterizing the essential spectrum of an operator $ T$ in $ {\mathcal{B}_n}(\Omega )$ for which $ \sigma (T) = \bar \Omega $ and the point spectrum of $ {T^ * }$ is empty. This class of operators forms an important part of $ {\mathcal{B}_n}(\Omega )$ denoted by $ {\mathcal{B}'_n}(\Omega )$. We use this characterization to give another proof of the result of Axler, Conway and McDonald on determining the essential spectrum of the Bergman operator.

Let $ {A_n}(G) = \left\{ {S:T = {S^ * }{\text{is}}\;{\text{in}}{{\mathcal{B}'}_n}({G^ * })} \right\}$. We also characterize the weighted shifts in $ {A_1}(G)$.

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Keywords: Essential spectrum, Bergman operator, weighted shift
Article copyright: © Copyright 1983 American Mathematical Society