On operators with closed analytic core

Miller, T. L.; Miller, V. G.; Neumann, Michael

doi:10.1007/bf02871857

Cited by 13 publications

(15 citation statements)

References 6 publications

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“…Hence, by the Hahn-Banach theorem, X T (int σ(T )) is dense in X. On the other hand, as noted above, σ loc (T ) = σ(T ), because T is the quotient of a decomposable operator; see [3] or [11]. Now the first assertion is immediate from Theorem 4.…”

Section: Resultsmentioning

confidence: 71%

“…As shown by Eschmeier and Prunaru [4], the localizable spectrum plays an important role in the theory of invariant subspaces; see also [3] and [11]. It is known that σ loc (T ) is a closed subset of σ(T ) and that σ loc (T ) contains the point spectrum and is included in the approximate point spectrum of T ; see Theorem 2 of [11]. Moreover, by the same result, σ loc (T ) = σ(T ) when T is decomposable or, more generally, the quotient of a decomposable operator by a closed invariant subspace, while σ loc (T ) may well be empty when T is the restriction of a decomposable operator.…”

Section: Resultsmentioning

confidence: 93%

Section: Resultsmentioning

confidence: 96%

“…Moreover, by the same result, σ loc (T ) = σ(T ) when T is decomposable or, more generally, the quotient of a decomposable operator by a closed invariant subspace, while σ loc (T ) may well be empty when T is the restriction of a decomposable operator. For instance, σ loc (T ) = ∅ if T is the unilateral right shift on p (N) for 1 ≤ p < ∞ or any pure isometry on a Hilbert space; see Example 3 of [11].…”

Section: Resultsmentioning

confidence: 99%

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Localizable spectrum and bounded local resolvent functions

Müller¹,

Neumann

2008

Arch. Math.

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Abstract. Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown that, for any measure with natural spectrum on a locally compact abelian group, the corresponding operator of convolution on the group algebra admits a non-zero bounded local resolvent function precisely when its spectrum has non-empty interior. Mathematics Subject Classification (2000). Primary 47A11; Secondary 43A25, 47A10, 47B40.

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Section: Resultsmentioning

confidence: 71%

Section: Resultsmentioning

confidence: 93%

Section: Resultsmentioning

confidence: 96%

Section: Resultsmentioning

confidence: 99%

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Localizable spectrum and bounded local resolvent functions

Müller¹,

Neumann

2008

Arch. Math.

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“…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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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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Normal Operators and their Generalizations

Aiena

2022

Trends in Mathematics

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On operators with closed analytic core

Cited by 13 publications

References 6 publications

Localizable spectrum and bounded local resolvent functions

Localizable spectrum and bounded local resolvent functions

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

Normal Operators and their Generalizations

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