The Beginning of the Spectral Theory of Nevanlinna’s Mapping fromTopological Space to Endomorphisms Algebra of Banach Space and its Applications

Misir B. Ragimov

Bulletin of the Malaysian Mathematical Society, Tome 26 (2003) no. 2

Résumé

The role of spectral theory of linear operators in qualitative theory of ordinary differential equations in Banach space is well known [1]. There wasn't the similar progress in theory of multidimensional differential equations because of abscence of satisfactory spectral theory, the linear mappings in the following from [8],[9]. (0.1) where E is real, F is complex Banach space and is Banach space of linear continuous operators, acting in F , and the values of operators commutate mutually. The interest for the study of spectral theory of such mappings is also explained by the demands of physics (spectral theory of commutative sets of self-adjoint operators (see momography of Berezansky [2])). On the investigation of mapping in form (0.1) and especially in the case when are unbounded operators, naturally arises the necessity of creation of spectral theory of some more general form of mappings, which include mappings (0.1) as a particular (private) case. While writing the spectral theory of such mappings we used Taylor's spectral theory [3,4] of mutually commutative set of operators. Main attention was paid to the application of such spectral theory to the solution of some questions of theory of multidimensional equations. Let's mark here, that in the case of finite dimensional spaces E and F (particularly F ) there is a big analogy with ordinary differential equations what cannot be said about infinite dimensional space F .