Define the right shift operator as $S: ℓ^2 \rightarrow ℓ^2, ||S||=1, S(x_1,x_2,...)=(0,x_1,x_2,...)$.
What is the adjoint $S^*$ and what are $SS^*$ and $S^*S$?
I know that $S^*$ is the left shift operator but how can I prove this? And what about the other two?