 Solution

Using $L=\Pi -I$ and the orthogonal projection identity ${\Pi }^{\ast }\Pi =\Pi$, we have $⟨\Pi f,f⟩=\Vert \Pi f{\Vert }_H^2$. The orthogonal decomposition $f=\Pi f+(I-\Pi )f$ then givesThe 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.