Is there an example of a differential operator $A(z)$ with parameter $z \in \mathbb{R}^d$ and Frechet derivative $A_z(z)$ such that $\mathrm{im}(A_z(z)) \subseteq \mathrm{ker}(A^T(z))$. Can this still be done when $d = \mathrm{dim(\mathrm{ker}(A(z)))}$? I am interested in operators satisfying these properties due to their connection with a decomposition of the scores of multivariate normal distributions $N(\mu, 1)$ with $\mu = A(z)x$.

1