Commutator formulas for gradient Ricci shrinker and their application to linear stability

In this paper we have found some commutator formulas between $di{v}_f,{\mathrm{\Delta }}_{f,L},di{v}_f^{†},{\mathrm{\Delta }}_f$ on closed orientable $GR{S}^{+}$ metrics (Gradient Ricci Shrinker), then with them we have generalized a Theorem of Cao and Zhu about necessary condition for linear stability of a $GR{S}^{+}$.