scholarly journals Weyl's Theorem for Algebraically (𝑝, π‘˜)-Quasihyponormal Operators

2006 β—½  
Vol 13 (2) β—½  
pp. 307-313
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space β—½  
Weyl’S Theorem β—½  
Hyponormal Operator β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Bounded Linear β—½  
Generalized Weyl’S Theorem β—½  
Weyl Spectrum β—½  
Hyponormal Operators β—½  
Isolated Eigenvalues

Abstract Let 𝐴 be a bounded linear operator acting on a Hilbert space 𝐻. The 𝐡-Weyl spectrum of 𝐴 is the set Οƒ 𝐡𝑀(𝐴) of all β‹‹ ∈ β„‚ such that 𝐴 – ⋋𝐼 is not a 𝐡-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 Οƒ 𝐡𝑀(𝐴) = Οƒ(𝐴) \ 𝐸(𝐴), and the 𝐡-Weyl spectrum Οƒ 𝐡𝑀(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, π‘˜)-quasihyponormal operator which includes an algebraically hyponormal operator.

scholarly journals Generalized Weyl's theorem and hyponormal operators

2004 β—½  
Vol 76 (2) β—½  
pp. 291-302 β—½  
Author(s):  
M. Berkani β—½  
A. Arroud
Keyword(s):  
Hilbert Space β—½  
Weyl’S Theorem β—½  
Hyponormal Operator β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Bounded Linear β—½  
Generalized Weyl’S Theorem β—½  
Weyl Spectrum β—½  
Hyponormal Operators β—½  
Isolated Eigenvalues

AbstractLet T be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of T is the set ΟƒBW(T) of all Ξ» ∈ Π‘such that T βˆ’ Ξ»I 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 ΟƒBW(T) = Οƒ(T)/E(T), and the B-Weyl spectrum ΟƒBW(T) of T satisfies the spectral mapping theorem. We also consider commuting finite rank perturbations of operators satisfying generalized Weyl's theorem.

scholarly journals A spectral mapping theorem for the Weyl spectrum

1996 β—½  
Vol 38 (1) β—½  
pp. 61-64 β—½  
Author(s):  
Woo Young Lee β—½  
Sang Hoon Lee
Keyword(s):  
Hilbert Space β—½  
Linear Operators β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Finite Multiplicity β—½  
Bounded Linear β—½  
Finite Dimensional β—½  
Weyl Spectrum β—½  
Isolated Points β—½  
Image Position

Suppose H is a Hilbert space and write β„’(H) for the set of all bounded linear operators on H. If T ∈ β„’(H) we write Οƒ(T) for the spectrum of T; Ο€0(T) for the set of eigenvalues of T; and Ο€00(T) for the isolated points of Οƒ(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 ∈ β„’(H) is said to be Fredholm if both Tβˆ’1(0) and T(H)βŠ₯ are finite dimensional. The index of a Fredholm operator T, denoted by index(T), is defined by

scholarly journals Weyl's theorem holds for p-hyponormal operators

1997 β—½  
Vol 39 (2) β—½  
pp. 217-220 β—½  
Author(s):  
Muneo Chō β—½  
Masuo Itoh β—½  
Satoru Ōshiro
Keyword(s):  
Point Spectrum β—½  
Complex Hilbert Space β—½  
Linear Operators β—½  
Weyl’S Theorem β—½  
Bounded Linear β—½  
Approximate Point Spectrum β—½  
Weyl Spectrum β—½  
Hyponormal Operators β—½  
Image Position β—½  
Isolated Eigenvalues

Let β„‹ be a complex Hilbert space and B(β„‹) the algebra of all bounded linear operators on β„‹. Let β„‹(β„‹) be the algebra of all compact operators of B(β„‹). For an operator T Ξ΅ B(β„‹), let Οƒ(T), Οƒp(T), σπ(T) and Ο€oo(T) denote the spectrum, the point spectrum, the approximate point spectrum and the set of all isolated eigenvalues of finite multiplicity of T, respectively. We denote the kernel and the range of an operator T by ker(T) and R(T), respectively. For a subset of β„‹, the norm closure of is denoted by . The Weyl spectrum Ο‰(T) of T Ξ΅ B(β„‹) is defined as the set

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

Filomat β—½  
2014 β—½  
Vol 28 (2) β—½  
pp. 411-419
Author(s):  
Young Han β—½  
Won Na
Keyword(s):  
Hilbert Space β—½  
Point Spectrum β—½  
Weyl’S Theorem β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Approximate Point Spectrum β—½  
Paranormal Operator β—½  
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 On quasi-M-hyponormal operators

Filomat β—½  
2011 β—½  
Vol 25 (1) β—½  
pp. 37-52 β—½  
Author(s):  
Young Han β—½  
Hee Son
Keyword(s):  
Positive Real Number β—½  
Point Spectrum β—½  
Hyponormal Operator β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Positive Real β—½  
Approximate Point Spectrum β—½  
Weyl Type Theorems β—½  
Weyl Spectrum β—½  
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.

Weyl's theorem for the square of operator and perturbations

2015 β—½  
Vol 17 (05) β—½  
pp. 1450042
Author(s):  
Weijuan Shi β—½  
Xiaohong Cao
Keyword(s):  
Hilbert Space β—½  
Compact Operator β—½  
Complex Hilbert Space β—½  
Linear Operators β—½  
Weyl’S Theorem β—½  
Bounded Linear Operators β—½  
Bounded Linear β—½  
Infinite Dimensional β—½  
Weyl Spectrum β—½  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if Οƒ(T)\Οƒw(T) = Ο€00(T), where Οƒ(T) and Οƒw(T) denote the spectrum and the Weyl spectrum of T, respectively, Ο€00(T) = {Ξ» ∈ iso Οƒ(T) : 0 < dim N(T - Ξ»I) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

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

1998 β—½  
Vol 220 (2) β—½  
pp. 760-768 β—½  
Author(s):  
Jin-Chuan Hou β—½  
Xiu-Ling Zhang
Keyword(s):  
Weyl’S Theorem β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Weyl Spectrum

scholarly journals Spectral properties of square hyponormal operators

Filomat β—½  
2019 β—½  
Vol 33 (15) β—½  
pp. 4845-4854
Author(s):  
Muneo Chō β—½  
Dijana Mosic β—½  
Biljana Nacevska-Nastovska β—½  
Taiga Saito
Keyword(s):  
Hilbert Space β—½  
Linear Operator β—½  
Bounded Linear Operator β—½  
Spectral Properties β—½  
Complex Hilbert Space β—½  
Hyponormal Operator β—½  
Bounded Linear β—½  
Basic Properties β—½  
Hyponormal Operators β—½  
Eigen Values

In this paper, we introduce a square hyponormal operator as a bounded linear operator T on a complex Hilbert space H such that T2 is a hyponormal operator, and we investigate some basic properties of this operator. Under the hypothesis ?(T) ? (-?(T)) ? {0}, we study spectral properties of a square hyponormal operator. In particular, we show that if z and w are distinct eigen-values of T and x,y ? H are corresponding eigen-vectors, respectively, then ?x,y? = 0. Also, we define nth hyponormal operators and present some properties of this kind of operators.

On Quasi-Class A Operators

2013 β—½  
Vol 59 (1) β—½  
pp. 163-172
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space β—½  
Analytic Function β—½  
Complex Hilbert Space β—½  
Linear Operators β—½  
Weyl’S Theorem β—½  
Bounded Linear Operators β—½  
Bounded Linear β—½  
Infinite Dimensional β—½  
Class A β—½  
Generalized Weyl’S Theorem

Abstract Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A;B be operators in B(H). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

scholarly journals On totally paranormal operators

2002 β—½  
Vol 66 (3) β—½  
pp. 425-441 β—½  
Author(s):  
Christoph Schmoeger
Keyword(s):  
Banach Space β—½  
Linear Operator β—½  
Continuous Linear Operator β—½  
Complex Banach Space β—½  
Weyl’S Theorem β—½  
Continuous Linear β—½  
Spectral Mapping Theorem β—½  
Spectral Mapping β—½  
Operator Equations β—½  
Weyl Spectrum

A continuous linear operator on a complex Banach space is said to be paranormal if β€–Txβ€–2 ≀ β€–T2xβ€– β€–xβ€– for all x ∈ X. T is called totally paranormal if T–λ is paranormal for every Ξ» ∈ 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.

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