Notes on Real Analytic Functions and Classical Operators

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This is a paper version of my presentation at Winter School in Complex Analysis and Operator Theory, Valencia February 2010. A real analytic function (i.e., possibly complex valued analytic function of a real argument) is one of the most classical objects of analysis. The theory of the whole class of this functions (treated as a topological vector space) and linear operators on them has developed recently due to new functional analytic tools available. The aim of this course is to survey this development with special emphasis on composition, differential and convolution operators on the space of real analytic functions and to show how our knowledge on the space helps to understand these operators. A nice feature of the theory is that a lot of deep classical theorems of real and complex analysis will be relevant and they will find unexpected relations to functional analytic results. The course will consists of four lectures: 1. Operator relevant properties of the space of real analytic functions • Topology on A (Ω) and tools for study operators used in the course • Relation with the Cousin problem 2. Composition operators on the space of real analytic functions • The space of real analytic functions as an algebra • When it has a closed range, when it is a topological embedding • Relation with analytic/algebraic geometry • How little do we know about hypercyclicity? 3. Differential and convolution operators on the space of real analytic functions • Surjectivity • Relation with algebraic geometry, Fourier analysis and the additive Cousin problem • How little do we know about parameter dependence and solution operators of differential and convolution equations on A (Ω)? 4. Isomorphism of the spaces of real analytic functions • Isomorphic classification for spaces over compact manifolds • Relation with composition and convolution operators • How little do we know about isomorphic classification over non-compact manifolds? 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. We consider spaces of real analytic functions over real analytic manifolds (both compact and non-compact). 12000 Mathematics Subject Classification. Primary: 46E10, 46E25, 26E05. Secondary: 14P15, 31A99, 32C05, 32U05, 34A35, 34K06, 35B35, 35E99, 35R10, 44A35, 46A04, 46A13, 46A35, 46A63, 46F15, 46M18, 47A16, 47A80, 47B33.