Spectral Properties of Tensor Products of Linear Operatrs. II: The Approximate Point Spectrum and Kato Essential Spectrum

1978 ◽  
Vol 237 ◽  
pp. 223 ◽  
Author(s):  
Takashi Ichinose
Keyword(s):  
Essential Spectrum ◽  
Spectral Properties ◽  
Point Spectrum ◽  
Tensor Products ◽  
Approximate Point Spectrum

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 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 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 Stability of essential approximate point spectrum and essential defect spectrum of linear operator

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1983-1994
Author(s):  
Aymen Ammar ◽  
Mohammed Dhahri ◽  
Aref Jeribi
Keyword(s):  
Linear Operator ◽  
Measure Of Noncompactness ◽  
Point Spectrum ◽  
Fredholm Operators ◽  
Approximate Point Spectrum ◽  
Densely Defined

In the present paper, we use the notion of measure of noncompactness to give some results on Fredholm operators and we establish a fine description of the essential approximate point spectrum and the essential defect spectrum of a closed densely defined linear operator.

scholarly journals Subdivisions of the Spectra for the Operator D(r, 0, 0, s) over Certain Sequence Spaces

2016 ◽  
Vol 34 (1) ◽  
pp. 75-84 ◽  
Author(s):  
Avinoy Paul ◽  
Binod Chandra Tripathy
Keyword(s):  
Point Spectrum ◽  
Sequence Spaces ◽  
Approximate Point Spectrum

In this paper we have examined the approximate point spectrum, defect spectrum and compression spectrum of the operator D(r, 0, 0, s)on the sequence spaces c0, c, ℓp and bvp.

scholarly journals Numerical Range of Two Operators in Semi-Inner Product Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
N. K. Sahu ◽  
C. Nahak ◽  
S. Nanda
Keyword(s):  
Point Spectrum ◽  
Numerical Range ◽  
Linear Operators ◽  
Inner Product ◽  
Product Spaces ◽  
Inclusion Relation ◽  
Nonlinear Operators ◽  
Approximate Point Spectrum ◽  
Inner Product Spaces ◽  
Inclusion Relations

In this paper, the numerical range for two operators (both linear and nonlinear) have been studied in semi-inner product spaces. The inclusion relations between numerical range, approximate point spectrum, compression spectrum, eigenspectrum, and spectrum have been established for two linear operators. We also show the inclusion relation between approximate point spectrum and closure of the numerical range for two nonlinear operators. An approximation method for solving the operator equation involving two nonlinear operators is also established.

Investigating Some Spectral Properties of Two-Dimensional Hysteresis Play-Functionals

2001 ◽  
Vol 08 (04) ◽  
pp. 303-313
Author(s):  
Andreas Ruffing
Keyword(s):  
Nonlinear Analysis ◽  
Spectral Properties ◽  
Point Spectrum ◽  
Lipschitz Constant ◽  
Two Dimensional ◽  
Hysteresis Operators ◽  
Building Elements

Spectral properties of two-dimensional generating functionals are considered. They are associated with scalar hysteresis operators. These operators occur as building elements for models of hysteresis within nonlinear analysis. We calculate eigenvalues of the hysteresis play-functionals and investigate the structure of the corresponding eigenvectors. It turns out that the point spectrum reflects the regularizing property that hysteresis play-operators exhibit in general: Their only possible eigenvalues are attained in the interval [0,1), thus reflecting the Lipschitz constant less than 1 for the play-operators.

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