Kernels of Toeplitz operators
Andreas Hartmann (IMB), Mishko Mitkovski

TL;DR
This survey explores the kernels of Toeplitz operators, focusing on criteria for non-trivial kernels, their structure, and related applications in various mathematical fields.
Contribution
It reviews key results on the characterization and structure of Toeplitz operator kernels, highlighting developments since 2005 and their applications.
Findings
Criteria for non-trivial kernels discussed
Structural properties of kernels analyzed
Connections to completeness and spectral problems established
Abstract
Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we will focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and P{\'o}lya sequences. Second, if the kernel is non-trivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review…
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Taxonomy
TopicsHolomorphic and Operator Theory · Analytic and geometric function theory · Approximation Theory and Sequence Spaces
