Two-dimensional Toeplitz operators with measurable symbols. (Q1889509)

Let \(1<p<\infty\), let \({\mathcal K}_p\) (resp. \({\mathcal K}_p^{(2)}\)) be the ideal of all compact operators on the Hardy space \(H^p({\mathbb T})\) of the unit circle \({\mathbb T}\) (resp., on the Hardy space \(H^p({\mathbb T}^2)\) of the torus \({\mathbb T}^2={\mathbb T}\times{\mathbb T}\)), and let \(TO_p\) (resp., \(TO_p^{(2)}\)) be the Banach algebra of all Toeplitz operators on \(H^p({\mathbb T})\) (resp., \(H^p({\mathbb T}^2)\)). Tensor-product techniques allow us to study the invertibility in \(TO_p\widehat{\otimes} TO_p/{\mathcal K}_p^{(2)}\) by reducing the problem to the invertibility in \(TO_p\widehat{\otimes} TO_p/TO_p\widehat{\otimes}{\mathcal K}_p\) and \(TO_p\widehat{\otimes} TO_p/{\mathcal K}_p\widehat{\otimes} TO_p\), where \(\widehat{\otimes}\) denotes the closure of the tensor product in the norm of \(H^p({\mathbb T}^2)\). This approach goes back to Douglas, Howe, Pilidi, Duduchava and was then developed in Chapter~8 of \textit{A. Böttcher} and \textit{B. Silbermann} [``Analysis of Toeplitz operators'' (Springer-Verlag) (1990; Zbl 0732.47029)]. But this approach fails for the larger algebra \(TO_p^{(2)}\) because \({\mathcal K}_p\widehat{\otimes}TO_p\) and \(TO_p\widehat{\otimes}{\mathcal K}_p\) are not ideals in \(TO_p^{(2)}\). To overcome these difficulties, the author introduces two new ideals \({\mathcal J}_p^1\) and \({\mathcal J}_p^2\) such that \({\mathcal J}_p^1{\mathcal J}_p^2\subset{\mathcal K}_p^{(2)}\) and \({\mathcal J}_ p^2{\mathcal J}_p^1\subset{\mathcal K}_p^{(2)}\). The invertibility in one of the algebras \(TO_p^{(2)}/{\mathcal J}_p^1\), \(TO_p^{(2)}/{\mathcal J}_p^2\) can be studied with the help of the localization techniques on the other. The author proves the following main results. Suppose that \(a,b\in L^\infty({\mathbb T})\). (1) If there exist closed sets \(F_1,G_1,F_2,G_2\subset {\mathbb T}\) such that \(F_1\cap G_1=F_2\cap G_2=\emptyset\) and \(a\in C({\mathbb T}^2\setminus F_1\times F_2)\), \(b\in C({\mathbb T}^2\setminus G_1\times G_2)\), then the Fredholmness of \(T^{(2)}(a), T^{(2)}(b)\in TO_p^{(2)}\) implies the Fredholmness of \(T^{(2)}(ab)\). (2) If \(b\in C({\mathbb T}^2)\) and \(T^{(2)}(a)\in TO_p^{(2)}\) is Fredholm, then \(T^{(2)}(ab)\) is Fredholm if and only if \(b\neq 0\) on \({\mathbb T}^2\) and both partial indices of \(b\) are zero. (3) If \(a\) is \(p\)-sectorial, then \(T^{(2)}(a)\in TO_p^{(2)}\) is Fredholm if and only if both partial indices of \(a\) are zero.