scholarly journals A Note on Property(gb)and Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Qingping Zeng ◽  
Huaijie Zhong
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Bounded Linear ◽  
Approximate Point Spectrum ◽  
Weyl Spectrum ◽  
Nilpotent Operators ◽  
The Stability ◽  
Quasinilpotent Operators ◽  
Study Property

An operatorT∈ℬ(X)defined on a Banach spaceXsatisfies property(gb)if the complement in the approximate point spectrumσa(T)of the upper semi-B-Weyl spectrumσSBF+-(T)coincides with the setΠ(T)of all poles of the resolvent ofT. In this paper, we continue to study property(gb)and the stability of it, for a bounded linear operatorTacting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting withT. Two counterexamples show that property(gb)in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

PROPERTY (Bgw) AND PERTURBATIONS

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
M.H.M. Rashid ◽  
T. Prasad
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Space Operator ◽  
Approximate Point Spectrum ◽  
Finite Rank Operators ◽  
Riesz Operators ◽  
Nilpotent Operators ◽  
The Stability ◽  
Isolated Eigenvalues

AbstractA Banach space operator T satisfies property (Bgw) if the complement in the approximate point spectrum σa(T) of the semi-B-essential approximate point spectrum σSHF+-(T) coincides with the set of isolated eigenvalues of T of Unite multiplicity E°(T). We find conditions for Banach Space operator tosatfafy the property (Bgw). We also study the stability of property (Bgw) under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T.

scholarly journals Joint spectra of operators on Banach space

1986 ◽  
Vol 28 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Point Spectrum ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Approximate Point Spectrum ◽  
Definition Of ◽  
Unit Vectors

Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ″() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ″(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that∥(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

scholarly journals On new strong versions of Browder type theorems

2018 ◽  
Vol 16 (1) ◽  
pp. 289-297
Author(s):  
José Sanabria ◽  
Carlos Carpintero ◽  
Jorge Rodríguez ◽  
Ennis Rosas ◽  
Orlando García
Keyword(s):  
Banach Space ◽  
Spectral Properties ◽  
Point Spectrum ◽  
Approximate Point Spectrum ◽  
Weyl Spectrum

AbstractAn operator T acting on a Banach space X satisfies the property (UWΠ) if σa(T)∖ $\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $(T) = Π(T), where σa(T) is the approximate point spectrum of T, $\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $(T) is the upper semi-Weyl spectrum of T and Π(T) the set of all poles of T. In this paper we introduce and study two new spectral properties, namely (VΠ) and (VΠa), in connection with Browder type theorems introduced in [1], [2], [3] and [4]. Among other results, we have that T satisfies property (VΠ) if and only if T satisfies property (UWΠ) and σ(T) = σa(T).

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

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

Let ℋ be a complex Hilbert space and B(ℋ) the algebra of all bounded linear operators on ℋ. Let ℋ(ℋ) be the algebra of all compact operators of B(ℋ). For an operator T ε B(ℋ), let σ(T), σp(T), σπ(T) and πoo(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 ℋ, the norm closure of is denoted by . The Weyl spectrum ω(T) of T ε B(ℋ) is defined as the set

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
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 σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. 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 Approximate point spectrum and commuting compact perturbations

1986 ◽  
Vol 28 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Vladimir Rakočević
Keyword(s):  
Banach Space ◽  
Essential Spectrum ◽  
Point Spectrum ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Approximate Point Spectrum ◽  
Infinite Dimensional ◽  
Compact Perturbations ◽  
Image Position

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Setσem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

scholarly journals Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means

2019 ◽  
Vol 17 (1) ◽  
pp. 1703-1715 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Finite Rank ◽  
Necessary Conditions ◽  
Operator Ideal ◽  
Linear Operators ◽  
Approximation Numbers ◽  
Bounded Linear ◽  
Weighted Means ◽  
Generalized Cesáro Sequence Space

Abstract Let E be a generalized Cesáro sequence space defined by weighted means and by using s-numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components $$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$ of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.

scholarly journals Generic stability of the spectra for bounded linear operators on Banach spaces

1999 ◽  
Vol 12 (1) ◽  
pp. 31-33
Author(s):  
Luo Qun
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Residual Subset ◽  
Bounded Linear ◽  
Generic Stability ◽  
The Stability

In this paper, we study the stability of the spectra of bounded linear operators B(X) in a Banach space X, and obtain that their spectra are stable on a dense residual subset of B(X).

scholarly journals On the joint spectra of doubly commuting n-tuples of semi-normal operators

1985 ◽  
Vol 26 (1) ◽  
pp. 47-50 ◽  
Author(s):  
Muneo Chō ◽  
A. T. Dash
Keyword(s):  
Point Spectrum ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Approximate Point Spectrum ◽  
Joint Point ◽  
Normal Operators ◽  
Definition Of ◽  
Image Position ◽  
Joint Point Spectrum ◽  
Unit Vectors

Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that.A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such thatA point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ j ≤ n.

scholarly journals Properties (S) and (gS) for bounded linear operators

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1641-1652 ◽  
Author(s):  
M.H.M. Rashid
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Linear Operators ◽  
Finite Multiplicity ◽  
Bounded Linear ◽  
Weyl Type Theorems ◽  
Weyl Theorem ◽  
Polaroid Operator ◽  
Isolated Eigenvalues

An operator T acting on a Banach space X obeys property (R) if ?0a(T) = E0(T), where ?0a(T) is the set of all left poles of T of finite rank and E0(T) is the set of all isolated eigenvalues of T of finite multiplicity. In this paper we introduce and study two new properties (S) and (gS) in connection with Weyl type theorems. Among other things, we prove that if T is a bounded linear operator acting on a Banach space, then T satisfies property (R) if and only if T satisfies property (S) and ?0(T) = ?0a(T), where ?0(T) is the set of poles of finite rank. Also we show if T satisfies Weyl theorem, then T satisfies property (S). Analogous results for property (gS) are given. Moreover, these properties are also studied in the frame of polaroid operator.

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