Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.

中文翻译：

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双截断 Toeplitz 算子的可逆性、Fredholmness 和核

介绍和研究了在两个（最终不同的）模型空间的正交补之间起作用的非对称对偶截断 Toeplitz 算子。在 $L^2(\mathbb T) \oplus L^2(\mathbb T)$ 上扩展成对运算符后，它们被证明是等效的，并且如果它们的符号在 $L^\infty(\mathbb T) 中是可逆的$, 到带有反符号的非对称截断 Toeplitz 算子。还建立了与卡莱森电晕定理的关系。这些结果用于研究各类双截断 Toeplitz 算子的 Fredholmness、可逆性和谱。