Dual truncated Toeplitz operators on the orthogonal complement of the model space $$K_u^2(=H^2 \ominus uH^2)$$ with u nonconstant inner function are defined to be the compression of multiplication operators to the orthogonal complement of $$K_u^2$$ in $$L^2$$ . In this paper, we give a complete characterization of the commutant of dual truncated Toeplitz operator $$D_z$$ , and we even obtain the commutant of all dual truncated Toeplitz operators with bounded analytic symbols. Moreover, we describe the nontrival invariant subspaces of $$D_z$$ .

中文翻译：

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对偶截断 Toeplitz 算子的交换子空间和不变子空间

模型空间 $$K_u^2(=H^2 \ominus uH^2)$$ 与 u 非常量内函数的正交补上的双截断 Toeplitz 算子被定义为乘法算子对 $ 的正交补的压缩$K_u^2$$ 在 $$L^2$$ 中。在本文中，我们完整地刻画了对偶截断托普利兹算子$$D_z$$的换算子，甚至得到了所有带有界解析符号的对偶截断托普利兹算子的换算子。此外，我们描述了 $$D_z$$ 的非平凡不变子空间。