On Some Problems of Operator Theory and Complex Analysis

Abstract

In 1955 Kadison \cite{14} asked whether the analogue of the classical Burnside's theorem of the Linear Algebra holds in the infinite dimensional case. We use reproducing kernels method to solve the Kadison question. Namely, we prove that any proper weakly closed subalgebra $\mathcal{A}$ of the algebra $\mathcal{B}\left( H\right) $ of bounded linear operators on infinite dimensional complex Hilbert spaces $H$ has a nontrivial invariant subspace, i.e., $\mathcal{A}$ is a nontransitive algebra. This solves The Transitive Algebra Problem positively, and hence Hyperinvariant Subspace Problem and Invariant Subspace Problem are also solved positively. In this context, we also consider the celebrated Riemann Hypothesis of the theory of meromorphic functions and solve it in negative.