Functional analysis originated at the start of the twentieth century and is concerned with the structure of infinite dimensional vector spaces and the linear operators between them. Banach spaces are the fundamental objects of study. Since then, the subject has developed to solve differential and integral equations, particularly those that arise in mathematical physics and engineering. Quantum mechanics has motivated study of several topics such as  the spectral theory of Schroedinger and Dirac operators.

Algebras of operators on Hilbert space are of central importance in quantum mechanics, and are one topic of study at Lancaster. Specific aspects include algebraic quantum field theory and quantum groups. A related area is the representation theory of infinite groups as operators on Hilbert space.  

In applications to engineering, semigroups of linear operators are studied to solve problems in control theory of linear systems. This topic is related to spectral theory and to complex analysis.

In recent years, noncommutative probability has been developed to study quantum theory, and the group studies quantum probability and stochastic processes.

In the 1940s, Wigner developed random matrix theory to provide models for atomic nuclei. The scope of applications has widened to include wireless communications and random matrix theory is related to operator algebras via free probability. 

The Analysis group at Lancaster has strong links to other centres in the UK including the North British Functional Analysis Seminar, and Algebraic Quantum Field Theory in the UK. The group participates in several international collaborations and hosts visits by distinguished researchers to the UK. In particular, the group belongs to the Alliance Hubert Curien collaborative programme between UK and France on noncommutative probability, matrix analysis and quantum groups. The group contributes to the International Workshop on Operator Theory and its Applications, and Lancaster hosted IWOTA 2021.