The power series $\sum_{n=0}^\infty n!\,z^n$ and holomorphic solutions of some differential equations in a Banach space

Let $A$ be a bounded operator on a Banach space. A question about the existence of holomorphic solutions of the equation $z^2Aw'+g(z)=w$ is studied. Moreover, general properties of power series of the form $\sum_{n=0}^\infty c_nA^nz^n$, $c_n\in\mathbb C$ are considered.