Notes on real analytic functions and classical operators (Q2906462)

The article under review is the paper version of the course on the space \(\mathcal{A}(\Omega)\) of real analytic functions given at the Winter School in Complex Analysis and Operator Theory in Valencia, February 2010.NEWLINENEWLINEFrom the abstract: ``We explain main ideas behind the proofs of the results and provide plenty of open problems together with their motivation and background. We try to be reasonably self-contained to make lectures accessible to non-specialists and especially to young mathematicians entering the subject.''NEWLINENEWLINEThese notes consist of four lectures. Each of them starts with a short introduction and an outline of the content of the respective part. Lecture 1 deals with the (natural) topology of the space of real analytic functions over a real analytic manifold. Here, the difference between compact and non-compact ones can be observed. It turns out that one can define two (possibly different) topologies on the space of real analytic functions. However, they are equivalent which is shown in Theorem 1.27. Lecture 2 deals with the algebraic properties of \(\mathcal{A}(\Omega)\). The main result of this section describes when the composition operator has closed range and is open onto its range. The second part of this lecture deals with the dynamical behaviour of the composition operator. Lecture 3 is devoted to differential and convolution operators on \(\mathcal{A}(\Omega)\). In particular, the following problems are discussed: surjectivity, parameter dependence of solutions, right inverses. Lecture 4 focuses on the isomorphic classification of the spaces of real analytic functions. The cases of compact and non-compact manifolds are treated separately.NEWLINENEWLINEFor the entire collection see [Zbl 1232.30005].