Composition Operators on Spaces of Dirichlet Series and Algebrability in Banach Algebras

This project will investigate timely questions on composition operators acting on Banach spaces of Dirichlet series, and on hypercyclic algebras.

These topics are currently the subject of great mathematical interest, however fundamental questions remain unresolved in the respective fields.

The first objective is to advance the theory of Banach spaces of Dirichlet series by employing operator theoretic function theory to understand the topological structure of the set of composition operators acting on these spaces. An effective method of achieving this is to study approximation numbers of differences of composition operators.

This project will determine the rates of decay of approximation numbers of differences of composition operators acting on spaces of Dirichlet series, and characterise the linear combinations of composition operators acting on general Banach spaces of Dirichlet series.

The second objective is to investigate the algebraic structure contained in the set of hypercyclic vectors of multiplication operators acting on the Banach algebra of compact operators.

Hitherto work in this area has mostly been in the setting of Frchet algebras, so this project will advance the theory for Banach algebras, and ultimately to identify whether every Banach algebra supports a hypercyclic algebra.

Interest among the wider mathematical community in the findings of this project stems from its natural connections to some of the most important open questions in mathematics.

In particular the study of Hardy spaces of Dirichlet series is related to the Riemann zeta function, and the investigation of hypercyclic algebras has a natural connection to the Invariant Subspace Problem.

The scientific breakthroughs resulting from this project will also impact on a wide scientific community, where the results and methods will potentially find applications in dynamical systems, mathematical physics, physics, computer science and quantum information theory.