Kernels of Toeplitz operators

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 Pólya sequences. Second, if the kernel is nontrivial, 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 a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.

Weighted model spaces and Schmidt subspaces of Hankel operators

“…We were also inspired by works on the structure of nearly ‐ invariant subspaces and the kernels of Toeplitz operators; see for early works and for later developments and surveys of the subject. In a companion paper we consider in detail the interesting special case when the spectrum of is finite, solve an inverse spectral problem involving the parameters , and give an explicit description of the corresponding class of symbols.…”

Geometric significance of Toeplitz kernels

“…Now we consider the case of two inner functions with support z = 1. As a direct consequence of the results on Toeplitz kernels obtained by Makarov, Mitkovski and Poltoratski [23,25] (see also the survey [19]), one can show examples of the two inner functions of the aforementioned type such that their corresponding subspaces can or cannot be joined by a geodesic in Gr. These remarkable results were proved for Toeplitz operators in Hardy spaces of the upper-half plane (and other classes of functions).…”

“…In contrast to the invertibility problem for Toeplitz operators, little attention has been paid in the literature to the injectivity problem until recent years. Except for the works of [12,22], the problem remained untreated until the recent works [23,24,25] (see also the survey [19]). Apart from being an interesting problem in operator theory, in these latter articles there are relevant applications to harmonic analysis, complex analysis and mathematical physics.…”

Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

“…One may ask, in that case, what are the natural classes of symbols to consider and what properties do those operators possess. In particular, we would like to examine their kernels and study what properties are shared with kernels of bounded Toeplitz operators, which have attracted great interest for their rich structure and the information that they provide on the corresponding Toeplitz operators (see for instance the recent survey paper [11]).…”