scholarly journals Propertiesandfor Bounded Linear Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. H. M. Rashid
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Extension Property ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Single Valued Extension Property ◽  
Bounded Linear ◽  
Weyl Type Theorems

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

scholarly journals WEYL TYPE THEOREMS FOR FUNCTIONS OF OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 493-505 ◽  
Author(s):  
SEN ZHU ◽  
CHUN GUANG LI ◽  
TING TING ZHOU
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Compact Perturbations ◽  
Weyl Type Theorems ◽  
Necessary And Sufficient

AbstractA-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

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.

On the invertibility of length two elementary operators

2015 ◽  
Vol 30 ◽  
pp. 916-913
Author(s):  
Janko Bracic ◽  
Nadia Boudi
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Existence Of A Solution ◽  
Bounded Linear ◽  
Elementary Operators ◽  
Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

scholarly journals GROWTH CONDITIONS FOR OPERATORS WITH SMALLEST SPECTRUM

2014 ◽  
Vol 57 (3) ◽  
pp. 665-680
Author(s):  
H. S. MUSTAFAYEV
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Growth Conditions ◽  
Invertible Operator ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Formula Group ◽  
Image Position

AbstractLet A be an invertible operator on a complex Banach space X. For a given α ≥ 0, we define the class $\mathcal{D}$Aα(ℤ) (resp. $\mathcal{D}$Aα (ℤ+)) of all bounded linear operators T on X for which there exists a constant CT>0, such that $ \begin{equation*} \Vert A^{n}TA^{-n}\Vert \leq C_{T}\left( 1+\left\vert n\right\vert \right) ^{\alpha }, \end{equation*} $ for all n ∈ ℤ (resp. n∈ ℤ+). We present a complete description of the class $\mathcal{D}$Aα (ℤ) in the case when the spectrum of A is real or is a singleton. If T ∈ $\mathcal{D}$A(ℤ) (=$\mathcal{D}$A0(ℤ)), some estimates for the norm of AT-TA are obtained. Some results for the class $\mathcal{D}$Aα (ℤ+) are also given.

scholarly journals On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Aftab Khan ◽  
Gul Rahmat ◽  
Akbar Zada
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Uniform Exponential Stability ◽  
Discrete Semigroup ◽  
Exponentially Stable ◽  
Discrete Semigroups

We prove that a discrete semigroup𝕋={T(n):n∈ℤ+}of bounded linear operators acting on a complex Banach spaceXis uniformly exponentially stable if and only if, for eachx∈AP0(ℤ+,X), the sequencen↦∑k=0n‍T(n-k)x(k):ℤ+→Xbelongs toAP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.

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.

Representations and Divisibility of Operator Polynomials

1978 ◽  
Vol 30 (5) ◽  
pp. 1045-1069 ◽  
Author(s):  
I. Gohberg ◽  
P. Lancaster ◽  
L. Rodman
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

Let be a complex Banach space and the algebra of bounded linear operators on . In this paper we study functions from the complex numbers to of the form

Complex strict and uniform convexity and hyponormal operators

1984 ◽  
Vol 96 (3) ◽  
pp. 483-493 ◽  
Author(s):  
Kirsti Mattila
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Uniform Convexity ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Image Position

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. The (spatial) numerical range of an operator TεB(X) is defined as the setIf V(T) ⊂ ℝ, then T is called hermitian. More about numerical ranges may be found in [8] and [9].

On Meromorphic Operators, I

1967 ◽  
Vol 19 ◽  
pp. 723-736 ◽  
Author(s):  
S. R. Caradus
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Finite Multiplicity ◽  
Bounded Linear Operators ◽  
Zero Eigenvalue ◽  
Bounded Linear ◽  
Zero Points

If X is a complex Banach space and B(X) denotes the space of bounded linear operators on X, then the class of meromorphic operators consists of those T in B(X) such that the non-zero points of σ(T) are poles of the resolvent Rλ(T). If we also require that each non-zero eigenvalue of T have finite multiplicity, members of the class ⊆ so defined have been called operators of Riesz type. and have been studied in (2, 6, 7) and (1,4) respectively.

Sign in / Sign up

Export Citation Format

Share Document