Holomorphic (analytic) operators and vector-functions on complex Banach spaces

Abstract

In Chapter I we introduced the notions of δ-differentiability, $\mathcal{G}$-differentiability and $\mathcal{F}$-differentiability of an operator $F:\mathfrak{X} \to \mathfrak{Y}$ at a given point. For each of these notions we defined the classes of m-times differentiable operators, 1 ≤ m ≤ ∞, as well as the class of analytic operators. In the general case, each of these classes is a proper part of the previous one. However, when $\mathfrak{X}$ and $\mathfrak{Y}$ are complex spaces and F is differentiable on a nieghborhood of the given point, the above mentioned classes coincide. This gives the possibility to extend the well-established theory of complex-differentiable operators, a theory with meany deep results. (We will see below that in many cases the study of the Fréchet derivative of an operator at a point has not only local consequences, but also provides indications about the global features of the behavior of the given operator.) On the other hand, many operatorial equations in a real space can be studied by considering the complexification of the space and by extending the operator such that the given equation reduces to an equation in a complex space. In order to emphasize the important role of the complex spaces in the theory of differentiable operators, we will use in what follows, for the complex case, the term “holomorphic” instead of the term “differentiable”. The definitions of holomorphic operators, holomorphic vector-functions, and so on, will be given below.