Operator Theory - Involving Toeplitz Operators and Model Spaces.
Lead Research Organisation:
University of Leeds
Department Name: Pure Mathematics
Abstract
Context of research
Hilbert spaces of analytic functions are of importance, both in their own right, and for their applications in systems and control theory, signal processing, and PDEs. This project is concerned with the relations between Toeplitz operators (first developed about 100 years ago) and model spaces, which are subspaces of Hardy spaces invariant under the backward shift operator.
Following the work of Hayashi, Hitt, and Sarason in the 1980s, the structure of Toeplitz kernels (kernels of Toeplitz operators) is well understood, and this has been formulated using model spaces, which are examples of Toeplitz kernels.
About ten years ago, Sarason initiated a programme of research into truncated Toeplitz operators, although examples of these had been studied for over 50 years. These are operators on model spaces, analogous to Toeplitz operators, and here a complete theory is still lacking.
Aims and objectives
The overall aim of the project is to develop the theory of truncated Toeplitz operators (and, to some extent, truncated Hankel operators) in parallel to the theory of certain block-matricial Toeplitz operators, and to answer various currently unsolved questions concerning their kernels, invariant subspaces, and similar features. Other related operators on model spaces will be introduced and studied, as appropriate.
The objectives are
The first objective of this project is to build on recent results of Câmara and Partington linking truncated Toeplitz operators with block-matricial Toeplitz operators (a process called equivalence by extension), whose kernels have been described by Chalendar, Chevrot and Partington in work extending the Hayashi/Hitt/Sarason ideas. It is expected that the recently-developed notion of maximal vectors for Toeplitz kernels will also play a part.
The project's further objectives will lead the student into seeking analogues of near-invariance (a property possessed by Toeplitz kernels) for kernels of truncated Toeplitz operators, and to develop further the theory of similar operators on model spaces.
Potential applications and benefits
As a project in pure mathematics, this work is not primarily driven by industrial applications, although model spaces have appeared in both control theory (controllability) and signal processing (Paley-Wiener spaces). At this point the benefits outside mathematics are purely long-term.
Research Areas: Pure Mathematics, Mathematical Analysis
Qualification to be attained: Ph.D degree in Mathematics
Hilbert spaces of analytic functions are of importance, both in their own right, and for their applications in systems and control theory, signal processing, and PDEs. This project is concerned with the relations between Toeplitz operators (first developed about 100 years ago) and model spaces, which are subspaces of Hardy spaces invariant under the backward shift operator.
Following the work of Hayashi, Hitt, and Sarason in the 1980s, the structure of Toeplitz kernels (kernels of Toeplitz operators) is well understood, and this has been formulated using model spaces, which are examples of Toeplitz kernels.
About ten years ago, Sarason initiated a programme of research into truncated Toeplitz operators, although examples of these had been studied for over 50 years. These are operators on model spaces, analogous to Toeplitz operators, and here a complete theory is still lacking.
Aims and objectives
The overall aim of the project is to develop the theory of truncated Toeplitz operators (and, to some extent, truncated Hankel operators) in parallel to the theory of certain block-matricial Toeplitz operators, and to answer various currently unsolved questions concerning their kernels, invariant subspaces, and similar features. Other related operators on model spaces will be introduced and studied, as appropriate.
The objectives are
The first objective of this project is to build on recent results of Câmara and Partington linking truncated Toeplitz operators with block-matricial Toeplitz operators (a process called equivalence by extension), whose kernels have been described by Chalendar, Chevrot and Partington in work extending the Hayashi/Hitt/Sarason ideas. It is expected that the recently-developed notion of maximal vectors for Toeplitz kernels will also play a part.
The project's further objectives will lead the student into seeking analogues of near-invariance (a property possessed by Toeplitz kernels) for kernels of truncated Toeplitz operators, and to develop further the theory of similar operators on model spaces.
Potential applications and benefits
As a project in pure mathematics, this work is not primarily driven by industrial applications, although model spaces have appeared in both control theory (controllability) and signal processing (Paley-Wiener spaces). At this point the benefits outside mathematics are purely long-term.
Research Areas: Pure Mathematics, Mathematical Analysis
Qualification to be attained: Ph.D degree in Mathematics
Organisations
People |
ORCID iD |
| Jonathan Partington (Primary Supervisor) | |
| Ryan O'Loughlin (Student) |
Publications
O'Loughlin R
(2020)
Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators
in Complex Analysis and Operator Theory
O'Loughlin R
(2022)
Symbols of compact truncated Toeplitz operators
in Journal of Mathematical Analysis and Applications
O'Loughlin R
(2020)
Toeplitz kernels and the backward shift
in Journal of Mathematical Analysis and Applications
O'Loughlin R
(2022)
Matrix-Valued Truncated Toeplitz Operators: Unbounded Symbols, Kernels and Equivalence After Extension
in Integral Equations and Operator Theory
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509681/1 | 01/10/2016 | 30/09/2021 | |||
| 1972662 | Studentship | EP/N509681/1 | 01/10/2017 | 31/03/2021 | Ryan O'Loughlin |
| Description | Two publications have been accepted. One has appeared in the Journal of Mathematical Analysis and Applications, and the other has appeared in Complex Analysis and Operator Theory. |
| Exploitation Route | The papers will be of interest to anyone working with Toeplitz operators in Operator Theory. This area of mathematics has applications in signal processing and control theory. |
| Sectors | Electronics,Other |