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We distinguish a class of unbounded operators in L r , r ≥ 1, related to the self-adjoint operators in L 2 . For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in L r -spaces are applied. 2000 Mathematics Subject Classification: Primary 47A35, 60F15; Secondary 40G10, 47B40.doi:10.4064/sm179-1-5 fatcat:doxaiqlz7rctrcavwgvaxvz57y