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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.

We establish the short exact sequences associated with the algebras generated by dual truncated Toeplitz operators on the orthogonal complement of the model space $K^{2}_{u}$ , and discuss spectral properties of dual truncated Toeplitz operators.

In this paper, we study the product problem of Toeplitz operators on the Bergman space of the unit disk. We characterize when the product of two Toeplitz operators T f T g is a finite rank perturbation of another Toeplitz operator T h , with f , g bounded harmonic and h in C 2 class with invariant Laplacian in L 1 . As a consequence, we show that there is no nontrivial rank one perturbation. However, in the case rank m ≥ 2 , we construct an example that shows there are bounded harmonic functions f , g and h such that T f T g − T h has rank exactly m .

In Rudin (Function theory in polydiscs, Benjamin, New York 1969) constructed a submodule in $$H^{2}({\mathbb {D}}^2)$$ that contains no nontrivial bounded functions. This paper shows that this type of submodules is Hilbert–Schmidt by selecting a suitable power sequence. In an attempt to generalized Rudin’s submodules, the paper defines zero-based submodule and makes a preliminary study about its Hilbert–Schmidtness.