
Generic objects (GA1727844S)
from 01/01/2017
to 31/12/2019 investigator

Objectives:
An object can be called generic if it occurs typically, in the sense that its copies can be found in every residual set in an appropriate space of objects. The aim of the project is to study generic objects appearing in several areas of mathematics, finding new tools for constructing and detecting such objects, and exploring their combinatorial structure. Wellknown examples of generic objects in model theory are Fraisse limits. Generic structures occur also naturally in topology (e.g. the Cantor set) and various areas of mathematical analysis (e.g. generic Banach spaces). Cohen's settheoretic forcing offers a strong tool of constructing generic objects, usually needed for specific purposes and not related to Fraisse limits. One of our objectives is to explore links between the settheoretic method of forcing and modeltheoretic methods for constructing universal homogeneous structures.



Topological and geometrical properties of Banach spaces and operator algebras II (GA1700941S)
from 01/01/2017
to 31/12/2019 investigator

Objectives:
We wish to investigate the structure of Banach spaces, C*algebras and Jordan algebras and their relationship. Main topics include quantitative approach to Banach spaces, various methods of separable reduction, decompositions of Banach spaces to smaller subspaces, integral representation of affine Baire functions, descriptive properties of weak topologies, small sets in Banach spaces and Polish groups, universal spaces in various categories of Banach spaces, operators and their numerical ranges, structure of abelian subalgebras of a C*algebra, of associative subalgebras of a Jordan algebra and related structures, different types of order in operator algebras, representation of morphisms on various substructures of operator algebras, Bell's inequalities and quantum correlations. We wish to focus especially on those problems where the mentioned areas intersect each other and by solving them to contribute to clarification of connections among various areas of functional analysis.



Methods of function theory and Banach algebras in operator theory V. (GA1407880S)
from 01/01/2014
to 31/12/2016 main investigator

Objectives:
The objectives of the present research project are investigations concerning:
1. orbits of operators, linear dynamics, invariant subspaces;
2. operator theory in function spaces;
3. operator positivity in matrix theory.



Asymptotics of Operator Semigroups (AOS(318910))
from 01/11/2012
to 31/10/2016 main investigator

Programme type: FP7 Marie Curie Action "International Research Staff Exchange Scheme"
Objectives:
The theory of asymptotic behaviour of operator semigroups is a comparatively new field serving as a common denominator for many other areas of mathematics, such as for instance the theory of partial differential equations, complex analysis, harmonic analysis and topology.
The primary interest in the study of asymptotic properties of strongly continuous operator semigroups comes from the fact that such semigroups solve abstract Cauchy problems which are often models for various phenomena arising in natural sciences, engineering and economics.
Knowledge of the asymptotics of semigroups allows one to determine the character of longtime evolution of these phenomena.
Despite an obvious importance, the asymptotic theory of oneparameter strongly continuous operator semigroups was for a very long time a collection of scattered facts rather than an organized area of research. The interest increased in the 1980s and the theory has witnessed a dramatic development over the past thirty years. Still there is a number of notorious open problems that have been left open. These missing blocks prevent the theory from being complete, slow down the development of the theory and discourage specialists from related fields to engage into the theory.
The goal of the project is to give new impetus to the theory of asymptotic behaviour of operator semigroups. To this aim we plan to extend and unify various aspects of the asymptotic theory of operator semigroups: stability, hyperbolicity, rigidity, boundedness, relations to Fredholm property, to work out new methods and to solve several longstanding open problems thus giving the theory its final shape.
We intend to create an international forum that enables and promotes a multi and cross disciplinary exchange of ideas, methods and tools under the common umbrella of asymptotic theory of operator semigroups. Thus we expect that, moreover, a wide range of modern analysis will benefit from the project.



Nonlinear analysis in Banach spaces (7AMB12FR003)
from 01/01/2012
to 31/12/2013 investigator

Programme type: MOBILITY  France
Objectives:
The goal of this project is to contribute to a better understanding of the following topics:



Methods of function theory and Banach algebras in operator theory IV (201/09/0473)
from 01/01/2009
to 31/12/2014 main investigator

Objectives:
The objectives of the present research project are investigations concerning: 1. existence of invariant subspaces and subsets with given properties; 2. operator theory in spaces of holomorphic functions; 3. orbits of operators, hypercyclic and supercyclic vectors, semigroups of operators; 4. interpolation problems, operator models and commutant lifting theorem.



Spectral theory of linear operators and reflexivity (MEB090905)
from 01/01/2009
to 31/12/2010 main investigator

Programme type: MOBILITY
Objectives:
Study of questions of spectral theory of linear operators concerning the reflexivity.



Orbits, invariant subspaces and positivity in operator theory (IAA100190903)
from 01/01/2009
to 31/12/2013 investigator

Objectives:
The objectives of the research project are investigations in the following mutually interconnected areas:
1. orbits of operators, hypercyclic and supercyclic vectors;
2. existence of invariant subspaces and subsets;
3. operator positivity in matrix theory.



Methods of function theory and Banach algebras in operator theory III. (201/06/0128)
from 01/01/2006
to 31/12/2008 main investigator

The objectives of the present research project are investigations concerning: 1. existence of invariant subspaces and subsets with given properties; 2. operator theory in spaces of holomorphic functions; 3. orbits of operators, hypercyclic and supercyclic vectors, semigroups of operators; 4. interpolation problems, operator models and commutant lifting theorem.
