Spectral Inclusion Theorems

Douglas, Ronald G.; Eschmeier, Jörg

doi:10.1007/978-3-0348-0411-0_10

Cited by 4 publications

(3 citation statements)

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“…However, the cluster value property for finite tuples in H ∞ (D) is equivalent to the validity of the Corona Theorem for H ∞ (D). This equivalence is well known and follows, for instance, as a direct application of Theorem 1 from [5].…”

mentioning

confidence: 78%

“…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…”

Section: Preliminariesmentioning

confidence: 92%

“…It is well known (see for instance [5,Proposition 6]) that the Berezin transform of a compact operator K ∈ L(H 2 (σ)) vanishes on the boundary of D in the sense that lim…”

Section: Preliminariesmentioning

confidence: 99%

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

Eschmeier

2013

Studia Math.

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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 , . . . , T fm ) ∈ L(H 2 (σ)) m with symbols fi ∈ H ∞ +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.

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mentioning

confidence: 78%

“…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…”

Section: Preliminariesmentioning

confidence: 92%