A possible error in Villani's monograph “Hypocoercivity”

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $$((\cdot,\cdot))$$ as in (8.1). Then in the Theorem 40 below (page 64, with the choice $$N = M^3$$) this inner product is changed to (8.4), which is definitely different from (8.1). Also, even though we accept (8.4) rather than (8.1) as the definition of $$((\cdot,\cdot))$$, I don't see why (8.5) implies the advertised conclusion (which basically says the existence of some constant $$\lambda$$ such that $$((h,Lh)) \geq \lambda\,((h,h))$$ (please ignore the real part as I am only looking at the real case). (Background information: here $$L := A^*A + B$$ and $$B$$ is antisymmetric). I sometimes feel that celebrated mathematicians are so careless and they didn't pay great attention to details. Thanks for any help!

• For reference, in case its status changes, here is a link to your (as-of-yet) unanswered MO question: mathoverflow.net/questions/387120/… . Commented Apr 2, 2021 at 18:36
• How is (8.4) different from (8.1)? There is the specialisation $N = M^3$ that you pointed out and seems, at first sight, coherent with the computations. The only other thing I see is the sign before $b$, but reading the text, $a$, $b$ and $c$ seem arbitrary, so I don't see how it matters. Commented Apr 2, 2021 at 18:47
• @peterag The MO post is also attached with a bounty, but no one has answered that yet Commented Apr 2, 2021 at 18:59
• @D.Thomine If you look at the "b"-terms, in (8.1) with $N = M^3$ this becomes $2b\,\langle MAh,M^3Ch\rangle$, whereas the corresonding term in (8.4) is $-2b\langle M^2Ah,Ch\rangle$, they differ not only by a sign! Commented Apr 2, 2021 at 19:04
• I retract what I said, the sign of $b$ may be important. You are right about the missing $M^2$ term. Anyway, a solution may be to ask directly the author; he maintains an errata page for this book, so this would have the advantage of making the answer widely accessible. Commented Apr 6, 2021 at 5:31