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Analysis at Infinity: Integral Equations, Limit Operators and Beyond

Periodic Reporting for period 1 - AnalysisAtInfinity (Analysis at Infinity: Integral Equations, Limit Operators and Beyond)

Reporting period: 2019-06-01 to 2021-05-31

Differential equations, or more precisely boundary value problems, are used to model a large portion of physical phenomena in contemporary science. Solving these boundary value problems or a least proving the existence of a unique solution is therefore one of the most important tasks of contemporary mathematics. One of the most popular approaches is via potential theory where the (linear) differential equation is transformed to an integral equation. In case the integrand stays bounded, the resulting integral equation is well understood, but in many applications the integrand turns out to be unbounded. The purpose of this project was to investigate these so-called singular integral equations and the associated operators, and apply the results to concrete operators such as Toeplitz operators and double layer potentials (DLPs).

Our novel approach was to transfer limit operator methods, which originate from the study of infinite matrices, to the theory of integral operators. The main idea of limit operator theory is that an infinite matrix can only contain finite information in finite space. To get the whole picture, one needs to look "at infinity". To access the information at infinity one has to shift the matrix along the integers and take the limit at infinity. This idea does not have a straightforward generalization to operators on continuous domains like surfaces. Just to name an obvious issue: it is not always clear what "infinity" exactly means, especially if we are dealing with operators on bounded domains. However, we managed to find a way to generalize these ideas to integral operators by reinterpreting some of the ingredients of limit operator theory. For example, to address the previously mentioned issue, we interpreted the boundary of a domain as "infinity" or conversely, infinity is interpreted as the boundary of the integers. The main principles and assumptions of limit operator theory were formalized in an algebraic language involving analytic and geometric terms so that it can be applied to a variety of contexts.

In the second part of the project we obtained a variety of new results in the theory of Toeplitz, Hankel and Toeplitz+Hankel operators. Moreover, we are currently finalising a paper concerning DLP operators. Our original expectation was to disprove a long-standing conjecture regarding the spectral radius of the DLP operator by considering a domain with a peculiar type of singularity. However, our analysis showed that the conjecture still holds in this case and thus further evidence for the validity of the conjecture is obtained. Further investigations on this will be needed in the future.
We started our work with the generalization of the limit operator framework and obtained a satisfactory result relatively quickly, which is now published in the Journal of Mathematical Analysis and Applications. We then started working on applications such as compactness and boundedness problems of Toeplitz and Hankel operators, which led to another 3 publications in international journals. In the final part of the project we studied the spectral radius problem for the DLP operator and managed to confirm it in a new special case. To obtain this result we combined methods from operator theory, harmonic analysis and numerics. Due to the complexity of the subject and several repercussions related to the global pandemic, we were not quite able to finish the paper until the end of the project, but expect the work to be completed within a few months and then published in a leading numerical analysis journal.

At the IWOTA 2019 in Lisbon we organized a special session on integral operators and applications to gather the experts in the field and obtain an overview of the state of the art. In September 2019 we attended the specialised SQuaRE workshop on Toeplitz+Hankel matrices at the American Institute of Mathematics, which led to the collaboration with Torsten Ehrhardt in one of our publications. The workshop at Reading in May 2020, for which we obtained funding from the London Mathematical Society, had to be postponed until further notice because of the global pandemic. Instead, we organized an online workshop in October 2020 with more than 100 participants. Additionally, we plan to catch up on the in-person workshop as soon as possible. We will use this opportunity to present the final outcomes of this project. For the same reason the dissemination of our results at conferences and other events has been significantly limited. However, some of the results have already been presented at a few online seminars and we will continue the dissemination at future conferences and seminars including some upcoming conferences in August 2021.

In terms of personal development, the Marie Curie Fellow taught two modules at the University of Reading, supervised 6 Bachelor students and co-supervised one PhD student in analysis. Furthermore, he gave the annual sample lecture ("Reading Scholars") for prospective students in early 2020. The team also received funding for an undergraduate research project which is currently running with two third year students.
We expect our new limit operator methods to be used in many future publications as they have shown their effectiveness already in several different applications. In particular, we solved several open problems in the theory of Toeplitz and Hankel operators such as a compactness characterisation of Hankel operators or the boundedness problem of Toeplitz+Hankel matrices. We expect further results in these directions in the future, but we might also see applications in entirely different areas because of how general the framework is.

As for our most ambitious objective, the spectral radius problem, we were not able to resolve it during the project. Nevertheless, we think that our novel approach involving a combination of arguments from operator theory, harmonic analysis and numerics will have some impact in future work. Indeed, our proposed method worked, but it showed that the designated counterexample actually satisfies the conjecture. Further investigation will be needed to confirm whether a counterexample can be found in this way or whether our approach may be generalized in some way to prove the conjecture. As it stands, the conjecture is still widely open.
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