My solution concerning a problem about Ergodic Recurrence requires me to prove that $\|P_T 1_B\| > 0$.
Where $P_T$ is the projection onto the space $I := \{f \in L^2 : f \circ T = f\}$, $T$ is a measure preserving mapping (so for all $B$ measurable $\mu(T^{-1} B) = \mu(B)$) and $B$ is of positive measure.
Can someone hint my why $\|P_T 1_B\|$ should be strictly positive? If $1_B$ would be in $I$ then I would have that $T^{-1} B = B$ which is probably not the case.