Shadowing, hyperbolicity, and Aluthge transforms

We introduce the concept of \(\epsilon \)-homoclinic points for invertible linear operators on Banach spaces. We establish that an operator is hyperbolic if and only if it possesses the shadowing property without any nonzero \(\epsilon \)-homoclinic points (for some \(\epsilon >0\)). Additionally, we demonstrate that a linear operator on a Banach space X exhibiting the shadowing property is uniformly contracting, uniformly expanding, possesses a nontrivial invariant closed subspace, or has a dense set of bounded orbits in X. Moreover, we demonstrate the invariance of the set of generalized hyperbolic operators under \(\lambda \)-Aluthge transforms, where \(\lambda \in \left( 0, 1\right) \). Additionally, we establish that the Aluthge iterates of an invertible operator converge to a hyperbolic operator solely when the initial operator is hyperbolic. Finally, we prove that the Aluthge iterates of hyponormal shifted hyperbolic weighted shifts diverge and also the existence of hyperbolic weighted shifts with divergent Aluthge iterates.