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Abstract

Let $H^2(D)$ be the Hardy space on a bounded strictly pseudoconvex domain $D \subset \mathbb C^n$ with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with $H^{\infty }$-symbol, we show that a Toeplitz tuple $T_f = (T_{f_1}, \ldots , T_{f_m}) \in L(H^2(\sigma ))^m$ with symbols $f_i \in H^{\infty } + C$ is Fredholm if and only if the Poisson–Szegö 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.