Construct a sequence of unitary operator converges to unilateral shift

Question

Consider the Hilbert space $l^2(\mathbb{Z})$, and $A$ is the unilateral shift s.t. $A(a_1,a_2,...)=(0,a_1,a_2,...)$. How to construct a sequence of unitary operator converges to $A$ in strong operator topology?

Answer 1

In $l^2(\mathbb{Z})$ the unilateral shift is unitary, so I guess your question is about $l^2(\mathbb{N})$. In such a case, the sequence of unitaries $U_n$, $$ U_n(a_1,a_2,\ldots):=(a_{n+1},a_1,a_2,\ldots,a_n,a_{n+2},a_{n+3}) $$ clearly satisfy $U_n\to A$ strongly.