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 $ST$?

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


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