convergence of sequence in WOT, SOT & norm topology Let $S:\ell^2\rightarrow\ell^{2}$ be the forward shift operator i.e. $S(x_1,x_2,...)=(0,x_1,x_2,...)$ then I have to show that $S^n$ converges to $0$ in WOT but not ...--prophetes.ai

convergence of sequence in WOT, SOT & norm topology Let $S:\ell^2\rightarrow\ell^{2}$ be the forward shift operator i.e. $S(x_1,x_2,...)=(0,x_1,x_2,...)$ then I have to show that $S^n$ converges to $0$ in WOT but not in SOT and norm toplogy. WOT convergence can be seen as $\langle S^nx,y\rangle\rightarrow\langle0,y\rangle =0$ for all $ x,y \in \ell^2$ . Is it correct? I am also not able to prove that it doesn't converge in the other two topologies. Kindly Help!!! Thanks & Regards in advance.

If it doesn't converge strongly then it doesn't converge in norm (consider the contrapositive), so try to find a counterexample there. To show that $\\{S^n\\}$ does not converge strongly to $0$ consider the vector $e_1\in\ell^2$ defined by $e_1(n)=1$ if $n=1$ and $e_1(n)=0$ otherwise.