Let $A$ be a selfadjoint elliptic differential operator on a compact manifold. In mathematical physics and differential topology one often defines its determinant using the $\zeta$-function regularization which will be denoted below by $\det_\zeta(A)$.
QUESTION. Let $A,B$ be two commuting selfadjoint elliptic differential operators. Then their product $AB$ is also selfadjoint and elliptic. Is it true that $\det_\zeta(AB)=\det_\zeta(A)\det_\zeta(B)?$
This question is closely related to this one $\zeta$-function regularized determinants
