# Analytic operator

at a point $x_0$
An operator $A$, acting from one Banach space into another, that admits a representation of the form $$A(x_0+h) - Ax_0 = \sum_{k=1}^{\infty}C_kh^k,$$ where $C_k$ is a form of degree $k$ and the series converges uniformly in some ball $\|h\|<r$. An operator is called analytic in a domain $G$ if it is an analytic operator at all points of this domain. An analytic operator is infinitely differentiable. In the case of complex spaces, analyticity of an operator in a domain is a consequence of its differentiability (according to Gâteaux) at each point of this domain. Examples of analytic operators are Lyapunov's integro-power series, and the Hammerstein and Urysohn operators with smooth kernels on the space $\mathcal C$ of continuous functions.