GENERALIZED HYPERBOLICITY, STABILITY AND EXPANSIVITY FOR OPERATORS ON LOCALLY CONVEX SPACES

The paper states and proves that every generalized hyperbolic operator on a locally convex space has the finite shadowing property, and in sequentially complete spaces it has the strict periodic shadowing property (Theorem 2), using the topological direct sum X=M⊕N, the canonical projections, and geometric-decay estimates from (GH3). The candidate solution proves the same result via the same structural decomposition, forward/backward recursions on M and N, and geometric-series bounds; the periodic case is handled by Neumann-series/series constructions whose convergence is ensured by sequential completeness. The constants are chosen so that the controlling neighborhood U depends only on V and not on the chain length/period, exactly as in the paper. Hence, both are correct and essentially the same proof. See the paper’s Definition 1 of generalized hyperbolicity and Theorem 2 with its proof for finite and periodic shadowing, including the sequential-completeness step for the infinite series constructions .