On Some Problems of Operator Theory and Complex Analysis
Mubariz T. Garayev

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 of the algebra of bounded linear operators on infinite dimensional complex Hilbert spaces has a nontrivial invariant subspace, i.e., 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHolomorphic and Operator Theory · Algebraic and Geometric Analysis · Matrix Theory and Algorithms
