The essential spectrum of Toeplitz tuples with symbols in \(H^{\infty } + C\) (Q2866769)

Let \(D\subset\mathbb{C}^n\) be a bounded strictly pseudoconvex domain with a smooth boundary. Let \(H^2(D)\) denote the Hardy space over \(D\). Extending earlier results by McDonald on the unit ball, \textit{N. P. Jewell} [Stud. Math. 68, 25--34 (1980; Zbl 0435.47036)] showed that, for \(f\) in \(H^{\infty}+C\), the Toeplitz operator \(T_{f}\) over \(H^2(D)\) is Fredholm if and only if the Poisson-Szegő extension of \(f\) is bounded away from zero in a neighbourhood of the boundary \(\partial D\). A similar result also holds for Toeplitz operators acting on the Bergman space over \(D\).NEWLINENEWLINEThe paper under review studies the same problem for tuples of Toeplitz operators. More specifically, for any tuple \(f=(f_1,\dots,f_m)\) of functions in \(H^{\infty}+C\), one defines \(T_{f}=(T_{f_1},\dots,T_{f_m})\). It is well known that \(T_{f}\) is a tuple of essentially commuting operators on both the Hardy and Bergman spaces over \(D\). Using Koszul complexes and Taylor spectra, one defines the essential spectrum of the tuple \(T_{f}\), denoted by \(\sigma_{e}(T_{f})\). The main result of the paper is the spectral formula NEWLINE\[NEWLINE\sigma_{e}(T_f) = \bigcap\big(\overline{f(U\cap D)}: \partial D \subset U \text{ open}\big).NEWLINE\]NEWLINE The result holds for the Hardy space as well as the Bergman space over \(D\). In the Hardy space case, the symbol \(f\) on the right-hand side is the Poisson-Szegő extension of \(f\).NEWLINENEWLINESince difficulties arise when extending Jewell's methods to tuples of Toeplitz operators, the author uses a different approach. This new approach makes use of Gelfand theory and a spectral mapping theorem obtained by \textit{M. Andersson} and \textit{S. Sandberg} [Stud. Math. 154, No. 3, 223--231 (2003; Zbl 1029.47013)].