Shadowing, Generalized hyperbolic and Aluthge transforms

TL;DR

This paper explores the properties of hyperbolic operators, shadowing, and Aluthge transforms on Banach spaces, providing new characterizations, invariance results, and convergence/divergence behaviors of these operators.

Contribution

It introduces the concept of r-homoclinic points, characterizes hyperbolic operators via shadowing and r-homoclinic points, and analyzes the invariance and convergence properties of Aluthge transforms.

Findings

Hyperbolic operators are characterized by shadowing and absence of nonzero r-homoclinic points.

The set of generalized hyperbolic operators is invariant under λ-Aluthge transforms.

Aluthge iterates of invertible hyperbolic operators converge, while those of shifted hyperbolic bilateral weighted shifts diverge.

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 $\lambda \in \left( 0,1 \right)$. 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.