= Solution
The <position operator> $X$ and <momentum operator> $P$ are <self-adjoint operators>, so the <adjoint operator> is
$$
\boxed{a_+=P+im\omega X.}
$$
Use the <canonical commutation relation> with the order of factors preserved:
$$
\begin{aligned}
a_+a_-&=(P+im\omega X)(P-im\omega X)\\
&=P^2+m^2\omega^2X^2+im\omega(XP-PX)\\
&=P^2+m^2\omega^2X^2-m\hbar\omega
=\boxed{2m\left(H-\frac12\hbar\omega\right)},\\
a_-a_+&=P^2+m^2\omega^2X^2-im\omega(XP-PX)\\
&=P^2+m^2\omega^2X^2+m\hbar\omega
=\boxed{2m\left(H+\frac12\hbar\omega\right)}.
\end{aligned}
$$
Subtracting the first product from the second proves
$$
\boxed{[a_-,a_+]=2m\hbar\omega I.}
$$
These <creation and annihilation operators> are unscaled and carry the paper's phase convention; their <commutator> is not the unit-normalized oscillator commutator.
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