Product of two Linear Operators

Question

Let $S(e_n)=e_{n+1}$ and $T(e_n)=e_{n+2}$ be two linear operators on the Hilbert space $l_2(N)$, the space of all sequences $\sum_{1}^\infty |a_k|^2 < \infty$, and $\{e_n\}, n=0,1,2,...$ is the standard orthonormal basis.

How do I find the formula for $ST$?

Answer 1

By the way, $T = S^2$, so $ST = S^3$, and we can generalize Adrian's comment as $$S^k(e_n) = e_{n+k}$$ for any $k \ge 0$.