Year: 2022
Type: preprint
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 $λ$-Aluthge transforms for every $λ\in \left( 0,1 \right)$. 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.
Source: arXiv (Cornell University)
Author: Tran, Linh T. T.
Institution: Chungnam National University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Chungnam National University
OpenAlex ID: W4297847349