Weyl’s Theorem and Perturbations

Oudghiri, Mourad

doi:10.1007/s00020-004-1342-4

Cited by 30 publications

(8 citation statements)

References 15 publications

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“…This generalizes, and brings under one umbrella, a number of results from [2 3], [7,8], [9,11,13,14], [16], [17], [25,26] and others. such that p(A) = 0; T ∈ B(X) is (algebraically) polynomially PS, T ∈ P(PS), if there exists a non-constant polynomial q(.)…”

Section: Introductionsupporting

confidence: 53%

Polaroid operators, SVEP and perturbed Browder, Weyl theorems

Duggal¹

2007

Rend. Circ. Mat. Palermo

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A Banach space operator T ∈ B(X) is polaroid, T ∈ P, if the isolated points of the spectrum of T are poles of the resolvent of T . Let PS denote the class of operators in P which have have SVEP, the single-valued extension property. It is proved that if T is polynomially PS and A ∈ B(X) is an algebraic operator which commutes with T , then f (T + A) satisfies Weyl's theorem and f (T * + A * ) satisfies a-Weyl's theorem for every f which is holomorphic on a neighbourhood of σ (T + A). (2000). Primary 47B47, 47A10, 47A11. Key words and phrases. Banach space, polaroid operator, single valued extension property, algebraic operator, polynomial operator, orthogonal subspaces, Browder's theorem, Weyl's theorem. 318 B. P. DUGGAL the classes consisting of hyponormal operators, log-hyponormal operators, p-hyponormal operators, M -hyponormal operators, w-hyponormal operators and totally * -paranormal Hilbert space operators. Furthermore, generalized scalar and subscalar operators in B(X), and multipliers of commutative semisimple Banach algebras are PS operators. (We refer the interested reader to [1, pp. 174-176] and [25] for further information; also see Section 4.) Recall, [11,13], that an operator T ∈ B(X) is said to be totally hereditarily normaloid, T ∈ T H N , if every part, and also T −1 p for every invertible part T p , of T is normaloid. Here a part of an operator is its restriction to an invariant subspace, and an operator is normaloid if its spectral radius equals its norm. Our first observation, which applies in particular to the class of paranormal operators [20, p. 229], is the following. PROPOSITION 1.1. T H N ⊂ PS. Proof. Recall from Dunford and Schwartz [15, p 93] that a subspace M of X is orthogonal to a subspace N of X in the sense of Garret Birkhoff, M ⊥ N , if ||m|| ≤ ||m + n|| for all m ∈ M and n ∈ N . This asymmetric definition of orthogonality reduces to the ususal (standard) definition of orthogonality in the case in which X = H is a Hilbert space. Let us say, for convenience, that T ∈ B(X) satisfies the orthogonality property O if (T − α) −1 (0) ⊥ T −1 (0) and (T − α) −1 (0) ⊥ (T − β) −1 (0) for all nontrivial distinct α,β ∈ C which are in the point spectrum σ p (T ) of T . (We remark here that whereas the polaroid property is invariant under similarities, property O is not similarity invariant.) It is known, see [13, Propositions 2.1 and 2.5], that T H N operators are simply polaroid and satisfy property O. The following argument, which is similar to that of [14, Lemma 3.9] (also see [3]), proves that T H N operators have SVEP. AMS(MOS) subject classificationAssume to the contrary that T does not have SVEP at a point α 0 . Then there exists a disc D centered at α 0 and a non-trivial analytic function f : D −→ X such that f (λ) ∈ (T − λ) −1 (0) for all λ ∈ D. Let μ ∈ D be such that f (μ) = 0, and let τ ∈ D \ {μ} such that f (τ ) = 0. If μ, τ are both non-zero, then the (mutual) orthogonality of (T − μ) −1 (0) and (T − τ ) −1 (0) implies that 0 < || f (μ)|| ≤ || f (μ) − f (τ )|| and 0 < || f (τ )|...

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Section: Introductionsupporting

confidence: 53%

Polaroid operators, SVEP and perturbed Browder, Weyl theorems

Duggal¹

2007

Rend. Circ. Mat. Palermo

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“…The perturbation result of Theorem 2.3 holds also if we suppose that K n is finite-dimensional for some n ∈ N, see [13]. The following example shows that, in general, an analogous result to that of Theorem 2.3 does not hold for property (w), even within the class of a-isoloid operators.…”

Section: Theorem 22 (Seementioning

confidence: 69%

Property (w) and perturbations II

Aiena

2008

Journal of Mathematical Analysis and Applications

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This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) [683][684][685][686][687][688][689][690][691][692] concerning the stability of property (w), a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T . A counterexample shows that property (w) in general is not preserved under finite-dimensional perturbations commuting with T , also under the assumption that T is a-isoloid. Definitions and basic resultsProperty (w) is a variant of Weyl's theorem that has been introduced by V. Rakočević in [14] and studied in more recent papers [3,4]. To introduce all these concepts, let X denote an infinite-dimensional complex Banach space and let L(X) be the algebra of all bounded linear operators on X. For an operator T ∈ L(X) we shall denote by α(T ) the dimension of the kernel ker T , and, by β(T ) the codimension of the range T (X). We recall that T ∈ L(X) is said to be upper semi-Fredholm if α(T ) < ∞ and T (X) is closed, while T ∈ L(X) is said to be lower semi-Fredholm if β(T ) < ∞. Let Φ + (X) and Φ − (X) denote the class of all upper semi-Fredholm operators and the class of all lower semi-Fredholm operators, respectively.The upper semi-Weyl operators are defined as W + (X) := T ∈ Φ + (X): indT 0 , the lower semi-Weyl operators are defined as W − (X) := T ∈ Φ − (X): indT 0 . The set of Weyl operators is defined by W (X) := W + (X) ∩ W − (X) = T ∈ Φ(X): indT = 0 . ✩ This research was supported by Fondi ex-60, 2005, Universitá di Palermo.

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“…Proof. Since Browder's theorem holds for T + R by Lemma 2.2 of [12], it suffices to show that π 0 (T + R) = E 0 a (T + R). If T − λ is invertible, then T + R − λ is a Fredholm, and hence λ ∈ E 0 a (T + R).…”

Section: Proof Start By Recalling That σ((mentioning

confidence: 99%

“…λ ∈ σ iso (S)) are eigenvalues of the operator. If S is finitely a-isoloid (i.e., if λ ∈ σ iso a (S) implies λ is a finite multiplicity eigenvalue of S), R ∈ L (X) is a Riesz operator which commutes with S, then S satisfies Weyl's theorem implies S + R satisfies Weyl's theorem [12,Theorem 2.7]. Given Banach space operators A ∈ L (X) and B ∈ L (Y), write…”

Section: Introductionmentioning

confidence: 99%

Passage of property (aw) from two operators to their tensor product

Rashid¹

2018

Kragujevac J Mathematics

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A Banach space operator S satisfies property (aw) if σ(S) \ σ w (S) = E 0 a (S), where E 0 a (S) is the set of all isolated point in the approximate point spectrum which are eigenvalues of finite multiplicity. Property (aw) does not transfer from operators A and B to their tensor product A ⊗ B, so we give necessary and/or sufficient conditions ensuring the passage of property (aw) from A and B to A ⊗ B. Perturbations by Riesz operators are considered.

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Weyl’s Theorem and Perturbations

Cited by 30 publications

References 15 publications

Polaroid operators, SVEP and perturbed Browder, Weyl theorems

Polaroid operators, SVEP and perturbed Browder, Weyl theorems

Property (w) and perturbations II

Passage of property (aw) from two operators to their tensor product

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