Suppose $T_i$ are invertible operators in $L^{2}(X)$ for X a Lebesgue Probability Space.
Is the following true?
1. If the $T_i$ converge weakly to $S$, then $S$ is not necessarily invertible.
2. If the $T_i$ converge weakly to $S$ and the $T_i$ commute with each other, then $S$ is invertible and in fact $T_i^{-1}$ converges weakly to $S^{-1}$.
I have already posted this question in ergodic theory section but I didn't have any answers so far. I restated it here in functional analytic point of view. ( Weak convergence of finite measure preserving transformations)