By the definition I have, for a map, $T$, to be self-adjoint, we have $\langle T v|w\rangle=\langle v|Tw\rangle$ for all $v,w\in V$ assuming $V$ is equipped with an inner product.
Suppose both $T,S$ are self-adjoint maps, then will that necessarily imply $ST,TS$ are self-adjoint as well? I tried to compute $\langle ST v|w\rangle=\langle Tv|Sw\rangle$ but what next?
Thanks!
Best Answer
$T^{t}$ is defined by $ \langle T^{t}x, y \rangle =\langle x, Ty \rangle$. Thus $T$ is self adjoint according to your definition iff $T^{t}=T$. We can show that $(ST)^{t}=T^{t}S^{t}$ in general. If $T$ and $S$ are self adjoint then $ST$ is self adjoint iff $(ST)^{t}=ST$ iff $T^{t}S^{t}=ST$ or $ST=TS$. So the answer is NO, the product $ST $ is self adjoint iff $ST=TS$.