Equality of Spectra of Quasi-Similar Hyponormal Operators

Clary, Stuart

doi:10.2307/2040373

Cited by 13 publications

(8 citation statements)

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“…By our assumption, there is an injective bounded linear operator with a dense range, I, from R 2 (Ω, ν) to R 2 (Ω, σ) such that IS ν = S σ I. Set u = I (1). Then I(f ) = uf, for each f ∈ A(Ω).…”

Section: Lemmamentioning

confidence: 99%

“…Though quasi-similarity is a weaker equivalence relation for operators, it is an interesting equivalence relation for the subnormal operators since quasi-similarity preserves spectrum [1] and essential spectrum [2] as well as some other properties for subnormal operators. So it would be desirable to describe the quasi-similar class of a given subnormal operator.…”

Section: Introductionmentioning

confidence: 99%

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On quasi-similarity of subnormal operators

Qiu

2007

SCI CHINA SER A

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For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R 2 (K, ν) denote the closure of Rat(K) in L 2 (ν) and let Sν denote the operator of multiplication by the inde-Suppose Ω is a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H 2 (Ω). Let σ denote a harmonic measure for Ω. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R 2 (Ω, σ). We show that for a given ν supported on Ω, Sν is quasi-similar to Sσ if and only if ν|∂Ω σ and log( dν dσ ) ∈ L 1 (σ). Our result extends a well-known result of Clary on the unit disk.

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Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On quasi-similarity of subnormal operators

Qiu

2007

SCI CHINA SER A

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“…There exist two hyponormal operators which are quasi-similar but not similar. In spite of the existence of such examples, Clary [8] showed that quasi-similar hyponormal operators have equal spectra.…”

Section: Corollary 28 If T ∈ L(h) Is W-hyponormal Then T Has Propementioning

confidence: 99%

Analysis of Non-normal Operators via Aluthge Transformation

Kimura

2004

Integr. equ. oper. theory

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Let T be a bounded linear operator on a complex Hilbert space H. In this paper, we show that T has Bishop's property (β) if and only if its Aluthge transformationT has property (β). As applications, we can obtain the following results. Every w-hyponormal operator has property (β). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chō's problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re σ(T ) = σ(Re T ) or not. (2000). Primary 47B20; Secondary 47A10, 47B40. Mathematics Subject Classification

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“…Two operators intertwined by a pair of quasi-affinities are said to be quasi-similar. Clary [5] showed that quasi-similar hyponormal operators have equal spectra.…”

Section: Introductionmentioning

confidence: 99%

Essential Spectral Inclusions for Operators on Banach Spaces

Miller

Neumann

2009

Mediterr. J. Math.

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This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T ) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T * do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop's property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context. Mathematics Subject Classification (2000). Primary 47A11; Secondary 47A10, 47A53, 47B20, 47B40.

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Equality of Spectra of Quasi-Similar Hyponormal Operators

Abstract: ABSTRACT.It is shown that quasi-similar hyponormal operators have equal spectra.

Cited by 13 publications

References 0 publications

On quasi-similarity of subnormal operators

On quasi-similarity of subnormal operators

Analysis of Non-normal Operators via Aluthge Transformation

Essential Spectral Inclusions for Operators on Banach Spaces

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