Shadowing lemmas for noninvertible operators and semigroups

We extend the scope of the shadowing lemma, initially proposed for generalized hyperbolic invertible operators on Banach spaces [3], to encompass both noninvertible operators and semigroups of operators. Moreover, we demonstrate that certain operators with the positive bounded shadowing property do not fall under the category of generalized hyperbolic operators (and similarly for semigroups). As a result, this provides a partial negative answer to a conjecture in [9].