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Details of Grant 

EPSRC Reference: EP/Y008375/1
Title: Toeplitz operators and their connections
Principal Investigator: Virtanen, Professor J
Other Investigators:
Researcher Co-Investigators:
Dr R O'Loughlin
Project Partners:
Department: Mathematics and Statistics
Organisation: University of Reading
Scheme: Standard Research
Starts: 01 March 2024 Ends: 28 February 2027 Value (£): 370,510
EPSRC Research Topic Classifications:
Mathematical Analysis Mathematical Physics
Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
16 Oct 2023 EPSRC Mathematical Sciences Prioritisation Panel October 2023 Announced
Summary on Grant Application Form
A Toeplitz matrix is constant along each diagonal parallel to the leading diagonal, and Toeplitz operators are infinite dimensional generalisations of Toeplitz matrices. A considerable variety of problems in Mathematics, Physics and Engineering can be expressed in terms of Toeplitz operators. Each Toeplitz operator is generated by a function, called the symbol, and the function-theoretic properties of the symbol can be used to determine properties of the Toeplitz operator. Truncated Toeplitz operators (TTOs) are function-theoretic generalisations of Toeplitz matrices, defined on certain spaces of analytic functions. The beauty of TTOs comes from the fact that large classes of perplexing operators can be modelled (via unitary equivalence) as TTOs. This includes convolution operators, Volterra type operators and several other classes of operators with a symmetry condition.

The asymptotic behaviors of the determinants of Toeplitz matrices as the size of the matrices tends to infinity are well understood when the derivative of the symbol has a certain degree of continuity. The Fisher-Hartwig asymptotics beautifully generalises this to the case when the symbol has singularities. In his groundbreaking work in the 1970s, Widom described the asymptotics of the determinants of block Toeplitz matrices with sufficiently nice symbols. However, in this block Toeplitz matrix case, the asymptotics are poorly understood when the symbol has singularities. This research project aims to describe the asymptotic behavior of the determinants of block Toeplitz matrices when the corresponding symbol has singularities. Complementary to the case of Toeplitz matrices, Böttcher has described the asymptotics of the determinants of a special class of block TTOs. The project will also aim to determine the asymptotics of the determinants of all block TTOs which have symbols with certain continuity properties. Another aim of the project is to apply the newly discovered determinant results to problems in other disciplines with a particular emphasis on quantum spin chain models, random tilings, and dimer models.

Just as polynomials can be applied to numbers, they can also be applied to matrices, and finding the magnitude (i.e. norm) of a linear mapping is one of the fundamental endeavors in Functional Analysis. Crouzeix's conjecture provides an optimal bound for the magnitude of a polynomial applied to a matrix. The conjecture was first posed in 2004, and to date has only been proved in special cases. This project aims to prove Crouzeix's conjecture, and it is expected that TTOs will play a key role in the resolution of this problem. Finding a bound for the norm of a matrix polynomial arises naturally in many situations, including analysing the stability of solutions of differential equations and in analysing the convergence rate of iterative linear system solvers such as the GMRES algorithm.

The characterisation of when a Toeplitz operator is continuous is the most famous cornerstone result in Toeplitz theory. Indeed, knowing when an operator is continuous is a useful and natural condition, with profound effects throughout Operator Theory and Harmonic Analysis. This project aims to prove the 1994 Berger-Coburn conjecture, which characterises the continuity of Toeplitz operators on the Fock space in terms of the heat equation, and to solve Sarason's 1994 product problem, which asks when the composition of Toeplitz operators is continuous. In addition to their importance in Operator Theory and Harmonic Analysis, answering these questions will develop new methods in emerging fields, such as Quantum Harmonic Analysis. Furthermore, these questions have connections to other disciplines such as mathematical physics, via quantization, and Machine Learning via Reproducing Kernel Hilbert Spaces.
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