Spectral mapping theorems for essential spectra

Gramsch, Bernhard; Lay, David C.

doi:10.1007/bf02052728

Cited by 79 publications

(39 citation statements)

References 14 publications

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“…The most general result in this direction [12] states that if f is an analytic function on an open set containing σ(T ), then w(f (T )) ⊆ f (w(T )) ; (0.1) but if T is hyponormal, then [17] …”

Section: The Subset Of σ(T ) That Is Stable Under Compact Perturbatiomentioning

confidence: 99%

Hyponormality and spectra of Toeplitz operators

Farenick

Lee

1996

Trans. Amer. Math. Soc.

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Abstract. This paper concerns algebraic and spectral properties of Toeplitz operators Tϕ, on the Hardy space H 2 (T), under certain assumptions concerning the symbols ϕ ∈ L ∞ (T). Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at Tϕ, for each quasicontinuous ϕ. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

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Section: The Subset Of σ(T ) That Is Stable Under Compact Perturbatiomentioning

confidence: 99%

Hyponormality and spectra of Toeplitz operators

Farenick

Lee

1996

Trans. Amer. Math. Soc.

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“…(2) See [6], Corollary 1, or [2], Theorem 1. (3) Let µ 0 ∈ σ 0 (f (T )); thus µ 0 is an isolated point in σ(f (T )) = f (σ(T )) and µ 0 ∈ F (f (T )).…”

Section: Preliminary Resultsmentioning

confidence: 99%

On the norm-closure of the class of hypercyclic operators

Schmoeger

1997

Ann. Polon. Math.

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Abstract. Let T be a bounded linear operator acting on a complex, separable, infinitedimensional Hilbert space and let f : D → C be an analytic function defined on an open set D ⊆ C which contains the spectrum of T . If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f (T ) is the limit of hypercyclic operators if and only if f (σ W (T )) ∪ {z ∈ C : |z| = 1} is connected, where σ W (T ) denotes the Weyl spectrum of T .

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“…Gramsch and D . Lay [2] have studied spectral mapping theorems for the essential spectra of an operator acting in a complex Banach space . In this note we introduce a notion of extended essential spectra for any closed linear operator with non-empty resolvent, which covers the aboye cases .…”

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confidence: 99%

On the spectral mapping theorem for essential spectra

Aleixandre¹

1985

Publicacions Matemàtiques

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.-B . Gramsch and D . Lay [2] have studied spectral mapping theorems for the essential spectra of an operator acting in a complex Banach space . In this note we introduce a notion of extended essential spectra for any closed linear operator with non-empty resolvent, which covers the aboye cases . Then, in this more general context, we are able to prove the spectral mapping theorems by means of a more unified approach based on a factorization of the operators provided by the DunfordTaylor calculus and well-known properties of products of operators present in Fredholm theory .

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Spectral mapping theorems for essential spectra

Cited by 79 publications

References 14 publications

Hyponormality and spectra of Toeplitz operators

Hyponormality and spectra of Toeplitz operators

On the norm-closure of the class of hypercyclic operators

On the spectral mapping theorem for essential spectra

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