scholarly journals On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

1998 ◽  
Vol 220 (2) ◽  
pp. 760-768 ◽  
Author(s):  
Jin-Chuan Hou ◽  
Xiu-Ling Zhang
Keyword(s):  
Weyl’S Theorem ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Weyl Spectrum

scholarly journals On totally paranormal operators

2002 ◽  
Vol 66 (3) ◽  
pp. 425-441 ◽  
Author(s):  
Christoph Schmoeger
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Continuous Linear Operator ◽  
Complex Banach Space ◽  
Weyl’S Theorem ◽  
Continuous Linear ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Operator Equations ◽  
Weyl Spectrum

A continuous linear operator on a complex Banach space is said to be paranormal if ‖Tx‖2 ≤ ‖T2x‖ ‖x‖ for all x ∈ X. T is called totally paranormal if T–λ is paranormal for every λ ∈ C. In this paper we investigate the class of totally paranormal operators. We shall see that Weyl's theorem holds for operators in this class. We also show that for totally paranormal operators the Weyl spectrum satisfies the spectral mapping theorem. In Section 5 of this paper we investigate the operator equations eT = eS and eTeS = eSeT for totally paranormal operators T and S.

scholarly journals Weyl’s theorem for algebraically quasi-paranormal operators

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 411-419
Author(s):  
Young Han ◽  
Won Na
Keyword(s):  
Hilbert Space ◽  
Point Spectrum ◽  
Weyl’S Theorem ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Approximate Point Spectrum ◽  
Paranormal Operator ◽  
Weyl Spectrum

Let T or T? be an algebraically quasi-paranormal operator acting on Hilbert space. We prove : (i) Weyl?s theorem holds for f (T) for every f ? H(?(T)); (ii) a-Browder?s theorem holds for f (S) for every S ? T and f ? H(?(S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.

scholarly journals Generalized Weyl's theorem and hyponormal operators

2004 ◽  
Vol 76 (2) ◽  
pp. 291-302 ◽  
Author(s):  
M. Berkani ◽  
A. Arroud
Keyword(s):  
Hilbert Space ◽  
Weyl’S Theorem ◽  
Hyponormal Operator ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Bounded Linear ◽  
Generalized Weyl’S Theorem ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Isolated Eigenvalues

AbstractLet T be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of T is the set σBW(T) of all λ ∈ Сsuch that T − λI is not a B-Fredholm operator of index 0. Let E(T) be the set of all isolated eigenvalues of T. The aim of this paper is to show that if T is a hyponormal operator, then T satisfies generalized Weyl's theorem σBW(T) = σ(T)/E(T), and the B-Weyl spectrum σBW(T) of T satisfies the spectral mapping theorem. We also consider commuting finite rank perturbations of operators satisfying generalized Weyl's theorem.

scholarly journals Weyl's Theorem for Algebraically (𝑝, 𝑘)-Quasihyponormal Operators

2006 ◽  
Vol 13 (2) ◽  
pp. 307-313
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Weyl’S Theorem ◽  
Hyponormal Operator ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Bounded Linear ◽  
Generalized Weyl’S Theorem ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Isolated Eigenvalues

Abstract Let 𝐴 be a bounded linear operator acting on a Hilbert space 𝐻. The 𝐵-Weyl spectrum of 𝐴 is the set σ 𝐵𝑤(𝐴) of all ⋋ ∈ ℂ such that 𝐴 – ⋋𝐼 is not a 𝐵-Fredholm operator of index 0. Let 𝐸(𝐴) be the set of all isolated eigenvalues of 𝐴. Recently, in [Berkani and Arroud, J. Aust. Math. Soc. 76: 291–302, 2004] the author showed that if 𝐴 is hyponormal, then 𝐴 satisfies the generalized Weyl's theorem σ 𝐵𝑤(𝐴) = σ(𝐴) \ 𝐸(𝐴), and the 𝐵-Weyl spectrum σ 𝐵𝑤(𝐴) of 𝐴 satisfies the spectral mapping theorem. Lee [Han, Proc. Amer. Math. Soc. 128: 2291–2296, 2000] showed that Weyl's theorem holds for algebraically hyponormal operators. In this paper the above results are generalized to an algebraically (𝑝, 𝑘)-quasihyponormal operator which includes an algebraically hyponormal operator.

scholarly journals On a class of operators

1998 ◽  
Vol 40 (2) ◽  
pp. 237-240
Author(s):  
Youngoh Yang
Keyword(s):  
Analytic Functions ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Spectrum Of An Operator ◽  
Weyl Spectrum ◽  
Equivalent Conditions

AbstractIn this paper we show that the Weyl spectrum of an operator of class W satisfies the spectral mapping theorem for analytic functions and give the equivalent conditions for an operator of the form normal + compact to be polynomially compact.

scholarly journals Some results on dominant operators

1998 ◽  
Vol 21 (2) ◽  
pp. 217-220 ◽  
Author(s):  
Youngoh Yang
Keyword(s):  
Analytic Functions ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Weyl Spectrum ◽  
Dominant Operator

We show that the Weyl spectrum of a dominant operator satisfies the spectral mapping theorem for analytic functions and then answer a question of Oberai.

scholarly journals A spectral mapping theorem for the Weyl spectrum

1996 ◽  
Vol 38 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Woo Young Lee ◽  
Sang Hoon Lee
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Finite Multiplicity ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Weyl Spectrum ◽  
Isolated Points ◽  
Image Position

Suppose H is a Hilbert space and write ℒ(H) for the set of all bounded linear operators on H. If T ∈ ℒ(H) we write σ(T) for the spectrum of T; π0(T) for the set of eigenvalues of T; and π00(T) for the isolated points of σ(T) that are eigenvalues of finite multiplicity. If K is a subset of C, we write iso K for the set of isolated points of K. An operator T ∈ ℒ(H) is said to be Fredholm if both T−1(0) and T(H)⊥ are finite dimensional. The index of a Fredholm operator T, denoted by index(T), is defined by

scholarly journals On quasi-M-hyponormal operators

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 37-52 ◽  
Author(s):  
Young Han ◽  
Hee Son
Keyword(s):  
Positive Real Number ◽  
Point Spectrum ◽  
Hyponormal Operator ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Positive Real ◽  
Approximate Point Spectrum ◽  
Weyl Type Theorems ◽  
Weyl Spectrum ◽  
Hyponormal Operators

An operator T is called quasi-M -hyponormal if there exists a positive real number M such that T ? (M 2 (T ??)? (T ??))T ? T ? (T ??)(T ??)? T for all ? ? C, which is a generalization of M -hyponormality. In this paper, we consider the local spectral properties for quasi-M -hyponormal operators and Weyl type theorems for algebraically quasi-M-hyponormal operators, respectively. It is also proved that if T is an algebraically quasi-M -hyponormal operator, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.

scholarly journals A spectral mapping theorem for perturbed Ornstein–Uhlenbeck operators on L2(Rd)

2015 ◽  
Vol 268 (9) ◽  
pp. 2479-2524 ◽  
Author(s):  
Roland Donninger ◽  
Birgit Schörkhuber
Keyword(s):  
Spectral Mapping Theorem ◽  
Spectral Mapping
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