Essential Spectral Inclusions for Operators on Banach Spaces |
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Authors: | T. Len Miller Vivien G. Miller Michael M. Neumann |
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Affiliation: | (1) Department of Mathematics and Statistics, Mississippi State University, Drawer MA, Mississippi State, MS 39762, USA |
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Abstract: | This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context. |
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Keywords: | KeywordHeading" >Mathematics Subject Classification (2000) Primary 47A11 Secondary 47A10 47A53 47B20 47B40 |
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