Abstract Differential Null-Equations

Let $A$ be a closed linear operator in a Banach space, and let $n\ge1$ be an integer. If the resolvent $(\lambda I-A)^{-1}$ is an entire function of $\lambda\in\mathbb{C}$ of order $<1/n$ or of order $1/n$ and minimal type, then the equation $d^nu(t)/dt^n=Au(t)$ has only the trivial solution $u(t)\equiv0$. An example for partial differential equations is given. Generalizations are indicated.