  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.
