In this paper, we investigate commuting dual truncated Toeplitz operators on the orthogonal complement of the model space $$K^{2}_{u}.$$ Let $$f,g \in L^{\infty },$$ if two dual truncated Toeplitz operators $$D_{f}$$ and $$D_{g}$$ commute, we obtain similar conditions of Brown–Halmos Theorem for Hardy-Toeplitz operators, that is, both f and g are analytic, or both f and g are co-analytic, or a nontrivial linear combination of f and g is constant. However, the first two conditions are not sufficient, one can easily construct two non-commuting dual truncated Toeplitz operators with analytic or co-analytic symbols. We prove that two bounded dual truncated Toeplitz operators $$D_{f}$$ and $$D_{g}$$ commute if and only if f, g, $${\bar{f}}(u-\lambda )$$ and $${\bar{g}}(u-\lambda )$$ all belong to $$H^{2}$$ for some constant $$\lambda ;$$ or $${\bar{f}},{\bar{g}}$$, $$f(u-\lambda )$$ and $$g(u-\lambda )$$ all belong to $$H^{2}$$ for some constant $$\lambda ;$$ or a nontrivial linear combination of f and g is constant.

中文翻译：

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对偶截断 Toeplitz 算子的 Brown-Halmos 型定理

在本文中，我们研究了模型空间 $$K^{2}_{u}.$$ 的正交补上的交换对偶截断 Toeplitz 算子让 $$f,g \in L^{\infty },$$如果两个对偶截断的托普利兹算子 $$D_{f}$$ 和 $$D_{g}$$ 交换，我们得到了 Hardy-Toeplitz 算子的 Brown–Halmos 定理的相似条件，即 f 和 g 都是解析的，或者 f 和 g 都是协解析的，或者 f 和 g 的非平凡线性组合是常数。然而，前两个条件是不充分的，可以很容易地构造出两个非对易对偶截断托普利兹算子与解析或共解析符号。我们证明两个有界对偶截断 Toeplitz 算子 $$D_{f}$$ 和 $$D_{g}$$ 交换当且仅当 f, g, $${\bar{f}}(u-\lambda ) $$ 和 $${\bar{g}}(u-\lambda )$$ 都属于 $$H^{2}$$ 对于某些常量 $$\lambda ;$$ 或 $${\bar{f }},