Sur la Theorie Spectrale Locale et Limite des Nilpotents

Мbekhta, Mostafa

doi:10.2307/2047902

Cited by 20 publications

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“…Remarque 1.1 in [6] shows that H 0 (T ) = X. Since H 0 (T ) ∩ K(T ) = {0}, by Proposition 4, we derive K(T ) = {0}.…”

Section: Christoph Schmoegermentioning

confidence: 72%

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A note on meromorphic operators

Schmoeger

2004

Proc. Amer. Math. Soc.

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Abstract. Let X be a complex Banach space and T a bounded linear operator on X. T is called meromorphic if the spectrum σ(T ) of T is a countable set, with 0 the only possible point of accumulation, such that all the nonzero points of σ(T ) are poles of (λI − T ) −1 . By means of the analytical core K(T ) we give a spectral theory of meromorphic operators. Our results are a generalization of some results obtained by Gong and Wang (2003).

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“…Remarque 1.1 in [6] shows that H 0 (T ) = X. Since H 0 (T ) ∩ K(T ) = {0}, by Proposition 4, we derive K(T ) = {0}.…”

Section: Christoph Schmoegermentioning

confidence: 72%

“…Proposition 1 (6) shows that 0 ∈ σ(T ). Proposition 2 implies then that 0 is the only possible isolated point of σ(T ).…”

Section: Theorem 4 Let T ∈ M(x) Thenmentioning

confidence: 97%

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A note on meromorphic operators

Schmoeger

2004

Proc. Amer. Math. Soc.

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“…Furthermore, T (K(T )) = K(T ) and if F is a closed subspace of X for which The next result may be found in Vrbová [20], or Mbekhta [16]; see also [14, Propositions 3.3.7 and 3.3.13].…”

Section: The Quasi-nilpotent Part Of An Operator and The Svepmentioning

confidence: 94%

Operators which have a closed quasi-nilpotent part

Aiena

Colasante

González

2002

Proc. Amer. Math. Soc.

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Abstract. We find several conditions for the quasi-nilpotent part of a bounded operator acting on a Banach space to be closed. Most of these conditions are established for semi-Fredholm operators or, more generally, for operators which admit a generalized Kato decomposition. For these operators the property of having a closed quasi-nilpotent part is related to the so-called single valued extension property. The quasi-nilpotent part of an operator and the SVEPThe single valued extension property was first introduced by Dunford An operator T ∈ L(X) is said to have the SVEP if T has the SVEP at every point λ ∈ C.

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“…It easily follows from the definition that K(T) is a linear subspace of X and that T(K(T)) = K(T); see [7]. The following lemma is needed.…”

Section: The Role Of the Analytic Corementioning

confidence: 99%

The Single-Valued Extension Property Is Not Preserved Under Sums and Products of Commuting Operators

Bourhim¹,

Miller²

2007

Glasgow Math. J.

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Abstract. We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.2000 Mathematics Subject Classification. Primary 47B37. Secondary 47A10, 47A11. Introduction.The problem of studying whether or not the local spectral properties such as the single-valued extension property, Dunford's condition (C), Bishop's property (β), the decomposition property (δ), or decomposability are preserved under sums and products of commuting operators has been considered by several authors, and remains an open problem; see for instance [3], [6], [9] and the references therein. Partial positive answers were obtained but only in certain special cases. In [9], T. L. Miller and M. M. Neumann showed that the sum and the product of two commuting operators with Dunford's condition (C) have the single-valued extension property. They also proved that the product of two commuting operators has this property provided that one of them is non-invertible and has fat local spectra.In this note, we show that the single-valued extension property is not preserved, in general, under the sums and products of commuting operators, and prove that the product of two commuting operators has this property provided that the intersection of their analytic cores is trivial. Our counter-examples are provided by tensor products of backward and forward unilateral weighted shifts, and our arguments and ideas are influenced by the ones given in [9].We now gather together some basic facts about the single-valued extension property and local spectrum. Our reference is the excellent book of K. Laursen and M. M. Neumann [6].Let X be a complex Banach space, and let L(X) be the algebra of bounded linear operators on X. For an operator T ∈ L(X), we denote as usual its spectrum and its

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Sur la Theorie Spectrale Locale et Limite des Nilpotents

Cited by 20 publications

References 0 publications

A note on meromorphic operators

A note on meromorphic operators

Operators which have a closed quasi-nilpotent part

The Single-Valued Extension Property Is Not Preserved Under Sums and Products of Commuting Operators

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