A theorem of Brown–Halmos type for dual truncated Toeplitz operators

Abstract

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.