# Product of two Linear Operators

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 $$ST$$?

• Wouldn't it just be $ST(e_n)=e_{n+3}?$ – Adrian Keister May 13 at 16:33
• The elements of $l_2(\Bbb N)$ are defined by $\sum_1^\infty \vert a_i \vert^2 = 0$, not $\sum_1^\infty a_i = 0$. – Robert Lewis May 13 at 17:19
• @RobertLewis Edited. Thank you. – A Slow Learner May 13 at 18:17
• It's $\sum_1^\infty \vert a_i \vert^2$! You still need to put the exponent 2 in! Cheers! – Robert Lewis May 13 at 18:18

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 $$k \ge 0$$.