Essential spectra of operators in the class $\mathcal {B}_n(\Omega )$
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- by Karim Seddighi PDF
- Proc. Amer. Math. Soc. 87 (1983), 453-458 Request permission
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 ${A_1}(G)$.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 453-458
- MSC: Primary 47A53; Secondary 47B38
- DOI: https://doi.org/10.1090/S0002-9939-1983-0684638-2
- MathSciNet review: 684638