The essential spectrum of holomorphic Toeplitz operators on H^pspaces

Abstract

Abstract. We compute the essential Taylor spectrum of a tuple of analytic Toeplitz, where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of T g provided that g −1 (0) is a compact subset of D.

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

“…In particular, it follows that lim z→z 0 H(z) = h(z 0 ) = 0. The above-cited results from [1] and [5] But then…”

“…Since Lemma 3.2 remains true with σ e (T f ) replaced by σ re (T f ) (see the proof of the lemma) and since the analytic spectral mapping formula used in the proof of Theorem 3.4 also holds for the right Taylor spectrum [6, Corollary 2.6.8], we obtain the following consequence. [1] for Toeplitz tuples with H ∞ -symbol by the corresponding spectral mapping formula for the Bergman space (Theorem 8.2.6 in [6]) and to replace the Poisson-Szegö transform by the Poisson-Bergman transform. All properties needed for the Poisson-Bergman integral can be found in [8].…”

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Let H 2 (D) be the Hardy space on a bounded strictly pseudoconvex domain D ⊂ C n with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with H ∞ -symbol, we show that a Toeplitz tuple T f = (T f 1 , . .

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