910346. Group of Linear Operators, Structure and Geometry of Banach Spaces

Department of Mathematical Analysis and Applied Mathematics
Faculty of Mathematical Science

This research project is included in the field of Mathematical Analysis, covering topics of Functional Analysis like Operator Theory and analytical techniques to study the geometry of Banach spaces. The main interests are focused around the lines:
1) Operator theory. In this framework, we focuses on the study of linear and bounded operators acting primarily in Hilbert spaces, the main issue to be addressed is to determine the structure of the given operator – for example, breaking it down in simpler operators, of simpler analysis, having as motivating goal the Invariant Subspace Problem. Operator Algebras are considered in this context, mainly in relation to understand structures of algebras generated by particular operators.
2) Geometry of Banach space and Banach lattices. While the study of geometry of Banach spaces is nowadays a classical subject, Banach lattices play an important role in order to determine the interplay about the properties of those operators acting on then and the underlined structure.
The IP Eva A. Gallardo and some research members are faculty members of the research institute Instituto de Ciencias Matemáticas ICMAT, depending on the National Research Council CSIC and the universities Autónoma de Madrid, Carlos III and Complutense .ICMAT has been awarded with the distinction of Severo Ochoa Center in Research & Development (by the Government of Spain).

The required infrastructure to carry over the research project is the corresponding to a research in Pure Mathematics; with acces to databases like MathScninet or ZentralblaTt , the library in Facultad de Matemáticas of Complutense University (one of the most complete in Europe) and access to computer facilities.

The scope of the project lays in the intersection between functional analysis, operator algebras and harmonic analysis. In particular, the main streams will be:
a) The use of noncommutative operator algebras, and their associated spaces of unbounded operators, to extend fundamental result of Harmonic analysis into the context of nonabelian groups. This requires bringing together tools from different areas of mathematics including von Neumann algebras, operator space theory, geometric group theory or the theory of abstract semigroups.
b) Well-posedness of some boundary value problems for a given differential operator on certain domain, mainly within the class of second order elliptic differential operator. At this regards, when a given class of elliptic system or scalar equation can be uniquely solved in some classes of domains and for some class on boundary data.
c) The study of the restriction problem in Harmonic Analysis, which asks whether there is a chance of having a meaningful restriction of the Fourier transform of a given function to curved submanifolds of Euclidean space, by considering probability measures supported on fractional dimensional compact sets.

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