Hi there I am trying to understand how the unilateral shift operator converges in the strong and weak topologies. Here is the question:
Suppose $S$ is a unilateral shift on $ℓ^2$ or $H^2$. Does the sequence $S^n$ converge in the strong and weak topologies to $0$? And does $(S^*)^n$ converge in the strong and weak topologies to $0$?
I know the unilateral shift operator is defined as $:ℓ^2⟶ℓ^2$ by $$(_1, _2, _3, … )= (0, _1, _2, _3, …).$$ But do I interpret $S^n$ and $(S^*)^n$? Could you please elaborate on how to approach this problem?
Hint: Whenever you want to determine the strong/weak convergence of a net of operators on $\ell^2$, determine how it behaves on the standard basis vectors.