Pseudospectral operators and the pointwise ergodic theorem

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We show that for a class of operators which properly contains the normal operators on L 2 {L_2} , \[ 1 n ∑ i = 0 n − 1 T i f → a . e . iff 1 2 n ∑ i = 0 2 n − 1 T i f → a . e . \frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.} \] This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.