Uniform ascent and descent of bounded operators

Grabiner, Sandy

doi:10.2969/jmsj/03420317

Cited by 122 publications

(93 citation statements)

References 17 publications

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“…This implies that λ 0 I − T has topological uniform descent, and since p(λ 0 I − T ) < ∞, it follows from Corollary 4.8 of [19] that λI − T is bounded below in a punctured disc centered at λ 0 .…”

Section: (Iv)⇒(i) Follows From Implication (1) (Iii)⇒(v)mentioning

confidence: 91%

“…By Theorem 1.8 we know that λ 0 I − T is quasi-Fredholm and hence has topological uniform descent. Since p(λ 0 I − T ) < ∞ it then follows, from Corollary 4.8 of [19] that λI − T is bounded below in a punctured disc centered at λ 0 , so λ / ∈ acc σ a (T ). To show the opposite inclusion, let λ 0 / ∈ σ usbf − (T ) ∪ acc σ a (T ).…”

Section: (Iv)⇒(i) Follows From Implication (1) (Iii)⇒(v)mentioning

confidence: 99%

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On generalized a-Browder's theorem

Aiena

Miller

2007

Studia Math.

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Abstract. We characterize the bounded linear operators T satisfying generalized a-Browder's theorem, or generalized a-Weyl's theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H 0 (λI − T ) as λ belongs to certain sets of C. In the last part we give a general framework in which generalized a-Weyl's theorem follows for several classes of operators.1. Preliminaries. Let L(X) denote the space of bounded linear operators on an infinite-dimensional complex Banach space X. For T ∈ L(X), denote by α(T ) the dimension of the kernel ker T , and by β(T ) the codimension of the range T (X). The operator T ∈ L(X) is called upper semiFredholm if α(T ) < ∞ and T (X) is closed, and lower semi-Fredholm if β(T ) < ∞. If T is either upper or lower semi-Fredholm then it is said to be semi-Fredholm; finally, T is a Fredholm operator if it is both upper and lower semi-Fredholm. If T ∈ L(X) is semi-Fredholm, then its index is defined by ind T := α(T ) − β(T ).For every T ∈ L(X) and a nonnegative integer n we shall denote by T [n] the restriction of T to T n (X) viewed as a map from T n (X) into itself (we set T

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Section: (Iv)⇒(i) Follows From Implication (1) (Iii)⇒(v)mentioning

confidence: 91%

Section: (Iv)⇒(i) Follows From Implication (1) (Iii)⇒(v)mentioning

confidence: 99%

On generalized a-Browder's theorem

Aiena

Miller

2007

Studia Math.

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“…The following definition describes the classes of operators we will study. These definitions were introduced by S. Grabiner [5].…”

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confidence: 99%

Weyl type theorem for operator matrices

Cao

2008

Studia Math.

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“…Operators with finite essential ascent play a significant role in [11], [12] and [17]. In [12], it was established that if T ∈ L (X) has finite essential ascent, then (1.3) R(T n ) is closed for some n > a e (T ) ⇔ R(T n ) is closed for all n ≥ a e (T ).…”

mentioning

confidence: 99%

“…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, Associated to an operator T on X we consider the non-increasing sequence ( [11]) c n (T ) = dim N(T n+1 )/N(T n ).…”

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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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Uniform ascent and descent of bounded operators

Cited by 122 publications

References 17 publications

On generalized a-Browder's theorem

On generalized a-Browder's theorem

Weyl type theorem for operator matrices

Ascent spectrum and essential ascent spectrum

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