Toeplitz operators on the disk with locally sectorial symbols (Q1316309)

The present note concentrates on Toeplitz operators whose symbols are locally sectorial in some sense. Assume, for example, \(\lambda\), \(\mu\), \(\nu\) are three distinct points on \(\mathbb{T}\) and \(a\in L^\infty\) is zero on the triangle \(\Delta\) made by \(\lambda\), \(\mu\), \(\nu\) and takes on constant values \(\alpha,\beta,\gamma\in\mathbb{C}\) on each of the three segments of \(\mathbb{D}\backslash\Delta\). We shall show that \(T(a)\) is Fredholm on \(A^2(\mathbb{D})\) if only the origin is not located on the periphery of the triangle spanned by \(\alpha\), \(\beta\), \(\gamma\). The approach employed here relies heavily on ideas from the circle-case, i.e., for Toeplitz operators on the Hardy space \(H^2(\mathbb{D})\), and it is to a large extent originated by the papers [\textit{B. Silbermann}, Integral Equations Oper. Theory 9, 706-738 (1986; Zbl 0609.47038)] and [the author, Semin. Anal. 1985/86, 1-17 (1986; Zbl 0634.47023)].