Strong convergence of rotation by 1/n operators

Question

let $ A_n: L^2([-\pi,\pi]) \rightarrow L^2([-\pi,\pi])$ be $ A_nf(t)=f(t-\frac{1}{n}) $.

Does $(A_n)$ converge in the strong operator topology?

Answer 1

It does converge in the strong operator topology. You can approximate $f$ in $L^2$ by a $2\pi$-periodic, continuous function $g$, and the strong convergence is clear for such a $g$.