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Shadowing, Generalized hyperbolic and Aluthge transforms - MaRDI portal

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Shadowing, Generalized hyperbolic and Aluthge transforms

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Publication:6410858

arXiv2209.07260MaRDI QIDQ6410858

Author name not available (Why is that?)

Publication date: 15 September 2022

Abstract: In this note, we introduce the notion of r-homoclinic points. We show that an operator on a Banach space is hyperbolic if and only if it is shadowing and has no nonzero r-homoclinic points. We also solve invariant subspace problem (ISP for brevity) for shadowing operators on Banach spaces. Afterwards, we verify that the set of generalized hyperbolic operators is invariant under lambda-Aluthge transforms for every lambdainleft(0,1ight). Next, the Aluthge iterates of invertible operators converge to hyperbolic operators only if the initial operators are hyperbolic. Finally, we prove that the Aluthge iterates of shifted hyperbolic bilateral weighted shifts diverge and that hyperbolic bilateral weighted shifts with divergent Aluthge iterates exist.





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