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 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 Spectrum of class absolute -*-k-paranormal operators for 0 ≤ k ≤ 1

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 671-678
Author(s):  
Changsen Yang ◽  
Junli Shen
Keyword(s):  
Point Spectrum ◽  
Open Neighborhood ◽  
Norm Inequality ◽  
Operator Inequality ◽  
Inclusion Relation ◽  
Approximate Point Spectrum ◽  
New Class ◽  
Joint Point ◽  
Weyl Spectrum ◽  
Joint Point Spectrum

In this paper, we shall introduce a new class absolute-*-k-paranormal operators given by a norm inequality and *-A(k) operator by operator inequality, we will discuss the inclusion relation of them. And we study spectral properties of class absolute-*-k-paranormal operators. We show that if T belongs to class absolute-*-k-paranormal operators, then its point spectrum and joint point spectrum are identical, its approximate point spectrum and joint approximate point spectrum are identical. Next as an application of them, for Weyl spectrum w(?) and essential approximate point spectrum ?ea, (?), we will show that if T or T*is absolute-*-k-paranormal for 0 ? k ? 1, then w(f (T)) = ? (w(T)), ?ea(? (T)) = ?(?ea(T)) for every ? ?? H(?(T)) where H(?(T)) denotes the set of all analytic functions on an open neighborhood of ?(T).

scholarly journals On a class of operators with normal Aluthge transformations

Filomat ◽  
2015 ◽  
Vol 29 (8) ◽  
pp. 1789-1794 ◽  
Author(s):  
Yousef Estaremi
Keyword(s):  
Large Class ◽  
Point Spectrum ◽  
Hölder Inequality ◽  
Weak Condition ◽  
Joint Point ◽  
Concrete Application ◽  
Joint Point Spectrum ◽  
Font Color

In this paper, we show that the generalized Aluthge transformations of a large class of operators (weighted conditional type operators) are normal. As a consequence, the operatorMwEMu is p-hyponormal if and only if it is normal, and under a weak condition, if MwEMu is normal, then the Holder inequality turn into equality for w; u. Also, we give some applications of p-hyponormal weighted conditional type operators, for instance, point spectrum and joint point spectrum of p-hyponormal weighted conditional type operators are equal. In the end, some examples are provided to illustrate concrete application of the main results of the paper. <br><br><font color="red"><b> This article has been retracted. Link to the retraction <u><a href="http://dx.doi.org/10.2298/FIL1601253E">10.2298/FIL1601253E</a><u></b></font>

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 On a class of operators with normal Aluthge transformations

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 969-975
Author(s):  
Yousef Estaremi
Keyword(s):  
Large Class ◽  
Point Spectrum ◽  
Hölder Inequality ◽  
Weak Condition ◽  
Joint Point ◽  
Concrete Application ◽  
Joint Point Spectrum

In this paper, we show that the generalized Aluthge transformations of a large class of operators (weighted conditional type operators) are normal. As a consequence, the operatorMwEMu is p-hyponormal if and only if it is normal, and under a weak condition, if MwEMu is normal, then the Holder inequality turn into equality for w; u. Also, we give some applications of p-hyponormal weighted conditional type operators, for instance, point spectrum and joint point spectrum of p-hyponormal weighted conditional type operators are equal. In the end, some examples are provided to illustrate concrete application of the main results of the paper.

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.

Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

2019 ◽  
Vol 12 (05) ◽  
pp. 1950084
Author(s):  
Anuradha Gupta ◽  
Ankit Kumar
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Spectral Properties ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

Let [Formula: see text] and [Formula: see text] be two bounded linear operators on a Banach space [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] and [Formula: see text], then [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have some common spectral properties. Drazin invertibility and polaroidness of these operators are also discussed. Cline’s formula for Drazin inverse in a ring with identity is also studied under the assumption that [Formula: see text] for some positive integer [Formula: see text].

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