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.

中文翻译：

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块 Toeplitz 算子的标量型核

结果表明，具有 $2\times 2$ 符号 $G$ 的 Toeplitz 算子的内核可以根据非常宽的类中的任何给定函数、其乘以 $G$ 的图像及其左逆来精确描述，如果后者存在。因此，在许多情况下，块 Toeplitz 算子的内核可以描述为标量复值函数空间与函数的固定列向量的乘积。这种核被称为标量类型，在本文中，它们在许多具体情况下进行了明确的研究和描述。给出了用于确定几种新符号类别的截断 Toeplitz 算子的核的应用。