Variation of the Point Spectrum under Compact Perturbations

Herrero, Domingo A.; Taylor, Thomas J.; Wang, Zong Y.

doi:10.1007/978-3-0348-5475-7_9

Cited by 21 publications

(4 citation statements)

References 16 publications

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“…Two of the main ingredients in the proof are ApostoΓs triangular representation and the decomposition of certain multiplication operators on L 2 -spaces. [Apo 2] is a good reference for the former, while [HTW,§3] is a good reference for the latter. What follows is a (very) brief synopsis of the salient features involved.…”

Section: σ(T)mentioning

confidence: 99%

Between the unitary and similarity orbits of normal operators

Guinand¹,

Marcoux²

1993

Pacific J. Math.

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has defined the (^ 4-Jf)-orbit of an operator T acting on a Hubert space iF to be (^ + Jr)(Γ) = {R~ιTR: R invertible of the form unitary plus compact}. In this paper, we characterize the norm closure in £ §(%?) of such an orbit in three cases: firstly, when T is normal; secondly when T is compact; and thirdly, when T is the unilateral shift. Some consequences of these characterizations are also explored.1. Introduction. Let %f be a complex, separable, infinite dimensional Hubert space and denote by 3 §(%?) the set of bounded linear operators acting on %?. As usual, 3£{%?) will denote the unique twosided ideal of compact operators. There are many interesting ways of partitioning the set 38{%?) into equivalence classes. We mention two in particular.Given T e 33{%T), we define the unitary orbit of T as W{T) = {U*TU: U e 33{%?) a unitary operator}. Then an operator A e %(T) if its action on %? is geometrically identical to that of T. Equivalently, one can think of A as T itself acting on an isomorphic copy of %r.Another much studied class is the similarity orbit of T e 3S[^), namely S?{T) = {S~ιTS: S e 33{fT) an invertible operator}. This notion of equivalence ignores the geometry of the Hubert space, and concentrates on the underlying vector space structure.In general, neither of these sets need be closed. This is in contrast to finite dimensional Hubert spaces, where %/(T) is always closed while S*(T) is closed if and only if T is similar to a normal matrix [Her 1, p. 14]. It is therefore interesting to describe the norm closure of these orbits, a program which for unitary orbits was undertaken by D. W. Hadwin One can also turn one's attention to the Calkin algebra s/ffi) = £%(^)/J?(β?) and consider both unitary and similarity orbits there. Indeed, one of the major results along these lines is the classification

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Section: σ(T)mentioning

confidence: 99%

Between the unitary and similarity orbits of normal operators

Guinand¹,

Marcoux²

1993

Pacific J. Math.

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“…The answer to both these problems is an emphatic "yes" (see [17,Theorems 1.4 and 1.5]. The argument used to prove these results ties up with the work of Herrero and his co-authors [11,12,13,4], Ji [15], and Zhu and Li [19]. A natural extension of this problem is the question of whether there exist compact operators K 0 , K ∈ B(H) such that (i)' A + K 0 is hereditarily polaroid and (ii)' A + K is not hereditarily polaroid.…”

Section: Introductionmentioning

confidence: 81%

Compact perturbations resulting in hereditarily polaroid operators

Duggal¹

2017

Bull. Belg. Math. Soc. Simon Stevin

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A Banach space operator A ∈ B(X ) is polaroid, A ∈ (P), if the isolated points of the spectrum σ(A) are poles of the operator; A is hereditarily polaroid, A ∈ (HP), if every restriction of A to a closed invariant subspace is polaroid.This, in answer to a question posed by Li and Zhou [17, Problem 5.5], proves the necessity of the condition Φis a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator K ∈ B(H) such that A + K ∈ (HP) is that Ω a (A) is connected.

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“…REMARK 5.3. As we have seen in the above proof, the result of Theorem 1.3 greatly depends on the work by D. Herrero [16] on perturbations of Hilbert space operators, and therefore, the result is established only in the setting of separable Hilbert spaces.…”

Section: Lemma 41 Let T ∈ B(x ) Then T ∈ (ω) If and Only Ifmentioning

confidence: 99%

Weyl Type Theorems for Functions of Operators

Zhu

Zhou

2012

Glasgow Math. J.

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Abstract.A-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f (T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ (T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.2000 Mathematics Subject Classification. Primary 47A53, 47A60; Secondary 47A10.1. Introduction. This paper is a continuation of a previous paper of the authors and Feng [17], where the stability of Weyl's theorem under holomorphic functional calculus is studied. A-Weyl's theorem and property (ω) as two variations of Weyl's theorem, which have been recently studied in [3, 4, 5], were introduced by Rakočević [21,22]. The purpose of this paper is to investigate a-Weyl's theorem and property (ω) for functions of operators on Banach spaces. A necessary and sufficient condition is given for an operator T to satisfy that f (T) obeys a-Weyl's theorem (property (ω)) for each function f analytic on some neighbourhood of σ (T).We first give some notations and terminologies. Throughout this paper, ‫ރ‬ and ‫ގ‬ denote the set of complex numbers and the set of natural numbers respectively. X will always denote a complex infinite dimensional Banach space. We let B(X ) denote the algebra of all bounded linear operators on X , and let K(X ) denote the ideal of compact operators in B(X ).Let T ∈ B(X ). We denote by σ (T) and σ p (T) the spectrum of T and the point spectrum of T respectively. Denote by n(T) and R(T) the kernel of T and the range

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Variation of the Point Spectrum under Compact Perturbations

Cited by 21 publications

References 16 publications

Between the unitary and similarity orbits of normal operators

Between the unitary and similarity orbits of normal operators

Compact perturbations resulting in hereditarily polaroid operators

Weyl Type Theorems for Functions of Operators

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