The essential spectrum of Toeplitz tuples with symbols in H∞ + C

Let H-2 (D) be the Hardy space on a bounded strictly pseudoconvex domain D subset of C-n with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with H-infinity-symbol, we show that a Toeplitz tuple T-f = (T-f1, ... ,T-fm) is an element of L(H-2(sigma))(m) with symbols f(i) is an element of H-infinity+C is Fredholm if and only if the Poisson-Szego extension of f is bounded away from zero near the boundary of D. Corresponding results are obtained for the case of Bergman spaces. Thus we extend results of McDonald (1977) and Jewell (1980) to systems of Toeplitz operators.