Functional analysis of infinite bounded operators

In mathematical research, it is well-known that many interesting linear operators, such as the derivative, are unbounded, i.e. they do not have any real Lipschitz constant. In this project, we suggest to instead consider such linear maps as bounded operators, but with a bound given by an infinite number in the ring of Colombeau generalized numbers. Linear maps with an (infinite) Lipschitz constant are called infinite bounded operators. The work packages we propose to develop are: 1. Theory of infinite bounded operators; 2. Applications to mathematical analysis and quantum mechanics; 3. Universal properties of generalized functions and of basic infinite bounded operators. The project aims at showing the flexibility of a non -Archimedean framework (i.e. of numbers which includes also infinitesimal and infinite constants), such as the Colombeau ring of generalized numbers, in strongly extending classical results of mathematical analysis using a simpler setting, and in showing important applications in solving singular differential equations and in quantum mechanics. The theory of infinite bounded operators represents a clear example of strong simplification due to a fundamental properties of infinitesimal and infinite numbers. It will surely stimulate further research of functional analysis in this framework. The possibility to use infinitesimal and infinite numbers in modeling physical systems is another important further feature of our approach. The primary researchers involved are: The applicant P. Giordano, Prof. M. Kunzinger and Prof. H. Vernaeve are the senior researchers working on this project. We also plan to employ two PhD candidates for the development of this project.

The project considered a non-Archimedean framework (i.e. a setting of new numbers which includes also infinitesimal and infinite constants), such as the Colombeau ring of generalized numbers, in strongly extending classical results of mathematical analysis using a simpler setting, and in showing important applications in solving singular differential equations and in quantum mechanics. We developed universal property of generalized functions (GF), graded Hilber spaces of GF, hyperfinite iterations of contractions of GF, Hahn-Banach theorem in spaces of GF, Riesz-Markov theorem in spaces of GF, applications to nonlinear mechanics, hyperfinite Fourier transform, PDE with non locally integrable terms. The possibility to use infinitesimal and infinite numbers in modeling physical systems is another important further feature of our approach. The primary researchers involved are: The applicant P. Giordano, Prof. M. Kunzinger and Prof. H. Vernaeve as senior researchers working on this project. We also employed four PhD candidates for the development of this project. Two of them successfully finished their PhD studies.