Abstract

We begin by showing that each generalized hyperbolic invertible operator in a Banach space possesses topological stability and the L-shadowing property. Next, we establish that any topologically stable invertible operator automatically exhibits the shadowing property, thereby ensuring the density of periodic points within its chain recurrent set. Afterwards, we establish that invertible operators possessing the L-shadowing property exhibit a form of hyperbolicity, characterized by the splitting of space into stable and unstable sets centered around the origin, and the nonwandering and chain recurrent sets coinciding with the closure of the homoclinic points. Moreover, the operator restricted to the closure of homoclinic points has the shadowing property. Lastly, we prove that an invertible operator of a Banach space is hyperbolic if and only if it has the L-shadowing property and no nonzero homoclinic points.

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