HYPOCOERCIVITY AND HYPOCONTRACTIVITY CONCEPTS FOR LINEAR DYNAMICAL SYSTEMS

The paper states (and reduces to) the short-time semigroup-norm expansion via the shifted hypocoercivity index: for B with Hermitian part BH and μ=λmin(BH), if mSHC(B)<∞ then ||e^{-Bt}||2 = e^{-μ t}(1 − c t^{2 mSHC(B)+1} + O(t^{2 mSHC(B)+2})) as t→0+, citing the semi-dissipative case and using the shift B↦B−μ I (Proposition 2.7 and Corollary 2.12) , with the factorization e^{-Bt}=e^{-μ t}e^{-B̃ t} explicitly noted and the SHC-index defined via the shift (Lemma 2.9/Definition 2.10) . The candidate solution proves the same statement directly: remove the skew part unitarily, analyze H(t)=e^{B̃S t}B̃H e^{-B̃S t} (even in t), derive the order of the first nonzero derivative of g_x(t)=||w(t)||^2, and obtain the exponent 2m+1 with a positive coefficient controlled by λmin(Tm)=λmin(Σ_{j=0}^m S^j H (S^H)^j)>0 (consistent with the paper’s equivalences for the HC-index) . The two arguments are logically compatible; the paper relies on earlier results, while the model gives a self-contained proof sketch with the same exponent and shift.