I am currently working through a set of lecture notes on operator theory.
For self - adjoint operators, I just showed that, if $B_1$ and $B_2 \in \mathcal{B}(\mathcal{H},)$ are self - adjoint then $B_1B_2$ is self - adjoint if and only if $B_1$ and $B_2$ commute. (Here, $\mathcal{H}$ denotes a Hilbert Space, and $\mathcal{B}(\mathcal{H})$ stands for the space of bounded operators defined on $\mathcal{H}$.
Now I am wondering - are there actually self - adjoint operators that do not commute?