= Solution
Using $L=\Pi-I$ and the <orthogonal projection> identity $\Pi^*\Pi=\Pi$, we have $\langle\Pi f,f\rangle=\|\Pi f\|_H^2$. The orthogonal decomposition $f=\Pi f+(I-\Pi)f$ then gives
$$
\boxed{\langle Lf,f\rangle=\|\Pi f\|_H^2-\|f\|_H^2=-\|f-\Pi f\|_H^2}.
$$
The sign is negative, as in the PDF; the converted TeX loses it at the last equality. This establishes the <exact spectral gap of normalized velocity relaxation>: on the zero-mass subspace $L=-I$, with gap one rather than the weaker half-gap estimate.
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