Problems in Modern Analysis

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The proposer will work on Operator Theory problems,problems related to invariant subspaces and tostability. The proposer has developed a new techniqueto construct invariant subspaces by using differentkinds of extremal vectors. This has, so far, led tonew and unified constructions of invariant subspaces for large classes of operators, including operatorscommuting with compact or normal operators. It has also led to strengthenings of so called Two SequencesTheorems, which are connected to invariant subspaces.There are several directions to refine and improve thistechniques and the proposer intends to do so. The study of stability under perturbations of cyclicity,supercyclicity or hypercyclicity is still in itsbeginning, both for operators and vectors. This isstrongly connected to the study of which properties arecarried by large resp. small classes of operators. Theproposer intends to carry these studies further.The study of operators and their invariant subspaces canbe seen as a generalization of linear algebra, where matrices and their eigenvectors are studied. Althoughthese are problems in "pure" mathematics, they areclosely connected to the many applications of linearalgebra to Physics, Biology, Genetics, Economics and other sciences as well as to the applications in industry. Operators can be seen as infinite matrices and are a suitabletool for the study of large and complex systems depending on anunbounded number of parameters.