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

Jörg Eschmeier¹

¹ Fachrichtung Mathematik Universität des Saarlandes D-66041 Saarbrücken, Germany

Studia Mathematica, Tome 219 (2013) no. 3, pp. 237-246

Résumé

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.

Détail

DOI : 10.4064/sm219-3-4

Keywords: hardy space bounded strictly pseudoconvex domain subset mathbb smooth boundary using gelfand theory spectral mapping theorem andersson sandberg toeplitz tuples infty symbol toeplitz tuple ldots sigma symbols infty fredholm only poisson szeg extension bounded away zero near boundary corresponding results obtained bergman spaces extend results mcdonald jewell systems toeplitz operators