(The) hyperinvariant subspace problem for quasinilpotent operators

Abstract

This dissertation concerns the yperinvariant subspace problem for quasinilpotent operators: Does every quasinilpotent operator on a complex infinite-dimensional separable Hilbert space have a nontrivial hyperinvariant subspace\,? We consider this problem by using the extremal vector technique and the model theory of quasinilpotent operators. Firstly, we focus on quasinilpotent operators in the class $(\mathcal{CRQ})$ with some constraints and then get partial solutions to the hyperinvariant subspace problem for quasinilpotent operators. Secondly, we introduce the notion of `stability' for extremal vectors and then prove that quasinilpotent operators with some stability conditions have nontrivial hyperinvariant subspaces. Finally, we also introduce the notion of `$c$-eigenvectors' and put it into the model theory of quasinilpotent operators to make improvements of the known results on the hyperinvariant subspace problem.