On Some Problems of Operator Theory and Complex Analysis

TL;DR

This paper proves that every proper weakly closed subalgebra of bounded operators on an infinite-dimensional Hilbert space has a nontrivial invariant subspace, solving several longstanding problems in operator theory.

Contribution

It uses reproducing kernels to positively resolve the Transitive Algebra Problem and related invariant subspace problems, and also provides a negative solution to the Riemann Hypothesis in meromorphic function theory.

Findings

Proper weakly closed subalgebras have nontrivial invariant subspaces

Transitive Algebra Problem is solved positively

Riemann Hypothesis in meromorphic functions is disproven

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.