INDIVIDUAL ERGODIC THEOREM FOR NON-CONTRACTIVE NORMAL
OPERATORS
Abstract: A condition implying the strong law of large numbers for trajectories of a normal
non-contractive operator is given. The condition has been described in terms of a spectral
measure, in the spirit of the well-known theorem of V. F. Gaposhkin. To embrace the
non-contractive operators we pass from the classical arithmetic (Cesŕro) means to the Borel
methods of summability.
1991 AMS Mathematics Subject Classification: 47A3S, 60F1S.
Key words and phrases: Strong law of large numbers, individual ergodic theorem,
normal operator, spectral measure, (non-)contractivity, Borel methods of summability,
Mittag-Leffler’s function, almost sure convergence.