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We characterize the solutions of the Poisson equation and the domain of itsassociated one-sided Hilbert transform for Ces\`aro bounded operators offractional order. The results obtained fairly generalize the corresponding onesfor power-bounded operators. In passing, we give an extension of the meanergodic theorem. Examples are given to illustrate the theory.

Poisson equation and discrete one-sided Hilbert transform for $(C,\alpha)$-bounded operators