Scalar-type kernels for block Toeplitz operators

TL;DR

This paper characterizes the kernels of certain block Toeplitz operators with 2x2 symbols, showing they can often be expressed as scalar function spaces multiplied by fixed vectors, with applications to truncated Toeplitz operators.

Contribution

It provides explicit descriptions of scalar-type kernels of block Toeplitz operators and extends these results to new classes of symbols for truncated Toeplitz operators.

Findings

Kernels can be expressed as scalar function spaces times fixed vectors.

Explicit descriptions of kernels are provided for many concrete cases.

Applications include determining kernels for new classes of truncated Toeplitz operators.

Abstract

It is shown that the kernel of a Toeplitz operator with $2\times 2$ symbol $G$ can be described exactly in terms of any given function in a very wide class, its image under multiplication by $G$, and their left inverses, if the latter exist. As a consequence, under many circumstances the kernel of a block Toeplitz operator may be described as the product of a space of scalar complex-valued functions by a fixed column vector of functions. Such kernels are said to be of scalar type, and in this paper they are studied and described explicitly in many concrete situations. Applications are given to the determination of kernels of truncated Toeplitz operators for several new classes of symbols.