A constructive view on ergodic theorems

Let T be a positive L1-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics. 1. The projection in L2 on the space N : = cl{x−T x: x∈L2} exists; 2. The sequence (T n)n∈N Cesàro-converges in the L2 norm; 3. The sequence (T n)n∈N Cesàro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.