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 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.

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 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 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

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 The spectrum of an R-homomorphism

1977 ◽  
Vol 23 (1) ◽  
pp. 42-45 ◽  
Author(s):  
A. W. Wickstead
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Point Spectrum ◽  
Positive Linear Operator ◽  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
Approximate Point Spectrum ◽  
Riesz Decomposition ◽  
The Riesz Decomposition Property

AbstractLet E be a real Banach space ordered by a closed, normal and generating cone. Suppose also that the order induced on E has the Riesz decomposition property. It is shown that if T:E → E is a positive linear operator with the property that y, z, a ∈ E with a ≧ Ty, Tz implies there is x ∈ E with x ≧ y, z and a ≧ Tx then the approximate point spectrum and spectrum of T are cyclic subsets of the complex plane. That is, if α = |α|γ lies in one of these sets then so does |α|γk for all integers k.

scholarly journals Spectrum of Quasi-Class (A,k) Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Xiaochun Li ◽  
Fugen Gao ◽  
Xiaochun Fang
Keyword(s):  
Positive Integer ◽  
Invariant Subspace ◽  
Spectral Properties ◽  
Point Spectrum ◽  
Approximate Point Spectrum ◽  
Common Generalization ◽  
Class A ◽  
Joint Point ◽  
Joint Point Spectrum

An operator T∈B(ℋ) is called quasi-class (A,k) if T∗k(|T2|−|T|2)Tk≥0 for a positive integer k, which is a common generalization of class A. In this paper, firstly we consider some spectral properties of quasi-class (A,k) operators; it is shown that if T is a quasi-class (A,k) operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of T are mutually orthogonal, and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam's theorems hold for class A operators. Particularly, we show that if T is a class A operator and either σ(|T|) or σ(|T∗|) is not connected, then T has a nontrivial invariant subspace.

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.

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