scholarly journals Spectral properties of tensor products of linear operators. II. The approximate point spectrum and Kato essential spectrum

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

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

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 Local spectral theory and spectral inclusions

1994 ◽  
Vol 36 (3) ◽  
pp. 331-343 ◽  
Author(s):  
Kjeld B. Laursen ◽  
Michael M. Neumann
Keyword(s):  
Classical Result ◽  
Point Spectrum ◽  
Linear Mapping ◽  
Linear Operators ◽  
Continuous Linear ◽  
Continuous Linear Mapping ◽  
Dense Range ◽  
Approximate Point Spectrum ◽  
Local Spectral Theory ◽  
Continuous Linear Operators

Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

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