Generalisation de la decomposition de kato aux opérateurs paranormaux et spectraux

Мbekhta, Mostafa

doi:10.1017/s0017089500006807

Cited by 124 publications

(82 citation statements)

References 7 publications

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“…The following two lemmas describe the basic properties of H 0 (T ) and K(T ); see [8,11]. Recall that T is said to be quasinilpotent when σ(T ) = {0}.…”

Section: Theorem 1 ([8 Théorème 16]) Let T ∈ B(x) and Let λ 0 ∈ σ(mentioning

confidence: 99%

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On the isolated points of the surjective spectrum of a bounded operator

Мbekhta¹,

Oudghiri²

2008

Proc. Amer. Math. Soc.

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Abstract. For a bounded operator T acting on a complex Banach space, we show that if T − λ is not surjective, then λ is an isolated point of the surjective spectrum σ su (T ) of T if and only if X = H 0 (T −λ)+K (T −λ), where H 0 (T ) is the quasinilpotent part of T and K(T ) is the analytic core for T . Moreover, we study the operators for which dim K(T ) < ∞. We show that for each of these operators T , there exists a finite set E consisting of Riesz points for T such that 0 ∈ σ(T ) \ E and σ(T ) \ E is connected, and derive some consequences.

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“…The following two lemmas describe the basic properties of H 0 (T ) and K(T ); see [8,11]. Recall that T is said to be quasinilpotent when σ(T ) = {0}.…”

Section: Theorem 1 ([8 Théorème 16]) Let T ∈ B(x) and Let λ 0 ∈ σ(mentioning

confidence: 99%

“…The quasinilpotent part H 0 (T ) and the analytic core K(T ) of T are subspaces of X that have been useful in the study of the spectral properties of operators [1,4,6,8,9,11,12,14]. For example, the following characterization of the isolated points of the spectrum σ(T ) was obtained in [8].…”

Section: Introductionmentioning

confidence: 99%

On the isolated points of the surjective spectrum of a bounded operator

Мbekhta¹,

Oudghiri²

2008

Proc. Amer. Math. Soc.

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“…This inclusion becomes an equality when the decreasing sequence {N(T ) ∩ R(T n ))} n is stationary (see [14]). So in particular, if T has finite essential ascent, Co(T ) = R ∞ (T ).…”

Section: T ) Is Finite If and Only If A E (T + F ) Is Finite If Thismentioning

confidence: 99%

“…Let us also recall the definition and the main properties of the analytic core and quasi-nilpotent part of T ∈ L (X). These subspaces have been introduced in [14] and deeply studied by Mbekhta in [14], [18] and [19].…”

Section: T ) Is Finite If and Only If A E (T + F ) Is Finite If Thismentioning

confidence: 99%

Ascent spectrum and essential ascent spectrum

2008

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Abstract. We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if σ e asc (T + F ) = σ e asc (T ) for every operator T commuting with F . The quasi-nilpotent part, the analytic core and the single-valued extension property are also analyzed for operators with finite essential ascent.1. Introduction. Throughout this paper X will be an infinite-dimensional complex Banach space. We will denote by L (X) the algebra of all operators on X, and by F (X) and K (X) its ideals of finite rank and compact operators on X, respectively. For an operator T ∈ L (X), write T * for its adjoint, N(T ) for its kernel and R(T ) for its range. Also, denote by σ(T ), σ ap (T ) and σ su (T ) its spectrum, approximate point spectrum and surjective spectrum, respectively. An operator T ∈ L (X) is upper semiFredholm (respectively lower semi-Fredholm) if R(T ) is closed and dim N(T ) (respectively codim R(T )) is finite. If T is upper or lower semi-Fredholm, then T is called semi-Fredholm. The index of such an operator is given by ind(T ) = dim N(T ) − codim R(T ), and when it is finite we say that T is Fredholm. Recall that for T ∈ L (X), the ascent, a(T ), and the descent, d(T ), are defined by

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“…(iii) An important case is obtained if we assume in the definition above that T |N is nilpotent. In this case T is said to be of Kato type; see [15]. Obviously, any semi-regular operator is of Kato type.…”

Section: Theorem 22 ([1]) For a Bounded Operator T On A Banach Spacmentioning

confidence: 99%

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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Generalisation de la decomposition de kato aux opérateurs paranormaux et spectraux

Abstract: Cette decomposition est connue sous le nom de decomposition de Kato de degre D. Les operateurs admettant une telle decomposition ont ete caracterises en 1978 par J. P. Labrousse

Cited by 124 publications

References 7 publications

On the isolated points of the surjective spectrum of a bounded operator

On the isolated points of the surjective spectrum of a bounded operator

Ascent spectrum and essential ascent spectrum

Operators which have a closed quasi-nilpotent part

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