Notes on real analytic functions and classical operators
type: Item
instance of: scholarly article
zbMATH Open document ID: 1267.46036
author: Paweł Domański
published in: Contemporary Mathematics
publication date: 2012-09-05T00:00:00Z
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 46-01
Mathematics Subject Classification ID: 00A07
zbMATH DE Number: 6077507
zbMATH Keywords: real analytic functions
zbMATH Keywords: DFS-spaces
zbMATH Keywords: PLS-spaces
zbMATH Keywords: additive Cousin problem
zbMATH Keywords: functor Proj
zbMATH Keywords: topological algebra of real analytic functions
zbMATH Keywords: closed ideals
zbMATH Keywords: maximal ideals
zbMATH Keywords: composition operators
zbMATH Keywords: closed range
zbMATH Keywords: analytic sets with extension property
zbMATH Keywords: semi-proper map
zbMATH Keywords: hypercyclic operator
zbMATH Keywords: power bounded operator
zbMATH Keywords: mean ergodic operator
zbMATH Keywords: Fourier transform
zbMATH Keywords: fundamental principle
zbMATH Keywords: convolution operator
zbMATH Keywords: linear partial differential operator with constant coefficients
zbMATH Keywords: Phragmén-Lindelöf principle
zbMATH Keywords: analytic parameter dependence of solutions of a partial differential operator or convolution operator
zbMATH Keywords: isomorphic classification
zbMATH Keywords: Schauder basis
zbMATH Keywords: finite type power series spaces
zbMATH Keywords: locally convex space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1090/conm/561/11108
OpenAlex ID: W2240844704
DOI: 10.1090/CONM/561/11108
title: Notes on real analytic functions and classical operators
review text: 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].
description: scientific article; zbMATH DE number 6077507