The factorization of operators which are close to the unit operator

2004 ◽  
pp. 156-211
Keyword(s):  
Unit Operator

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals Spectrum perturbations of compact operators in a Banach space

2019 ◽  
Vol 17 (1) ◽  
pp. 1025-1034
Author(s):  
Michael Gil’
Keyword(s):  
Banach Space ◽  
Integral Operators ◽  
Compact Operators ◽  
Constant Independent ◽  
Normed Ideal ◽  
Unit Operator

Abstract For an integer p ≥ 1, let Γp be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓp(.) and the property $$\begin{array}{} \displaystyle \sum_{k=1}^{\infty} |\lambda_k(A)|^p\le a_p N_{{\it\Gamma}_p}^p(A) \;\;(A\in {\it\Gamma}_p), \end{array}$$ where λk(A) (k = 1, 2, …) are the eigenvalues of A and ap is a constant independent of A. Let A, Ã ∈ Γp and $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A):= N_{{\it\Gamma}_p}(A-\tilde A) \;\exp\;\left[a_p b_p^p \;\left(1+\frac{1}2 (N_{{\it\Gamma}_p}(A+\tilde A) + N_{{\it\Gamma}_p}(A-\tilde A))\right)^p\right], \end{array}$$ where bp is the quasi-triangle constant in Γp. It is proved the following result: let I be the unit operator, I – Ap be boundedly invertible and $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A)\exp\;\left[\frac{a_pN^p_{{\it\Gamma}_p}(A) } {\psi_p(A)}\right] \lt 1, \end{array}$$ where ψp(A) = infk=1,2,… |1 – $\begin{array}{} \displaystyle \lambda_k^{p} \end{array}$(A)|. Then I – Ãp is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.

scholarly journals Multivariate and abstract approximation theory for Banach space valued functions

2017 ◽  
Vol 50 (1) ◽  
pp. 208-222 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Banach Space ◽  
Degree Of Approximation ◽  
Linear Operators ◽  
Approximation Properties ◽  
Uniform Estimates ◽  
Sharp Estimates ◽  
Differentiable Functions ◽  
Unit Operator ◽  
Fréchet Differentiable ◽  
High Degree

Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

scholarly journals Derivable mappings at unit operator on nest algebras

2007 ◽  
Vol 422 (2-3) ◽  
pp. 721-735 ◽  
Author(s):  
Jun Zhu ◽  
Changping Xiong
Keyword(s):  
Nest Algebras ◽  
Unit Operator

scholarly journals Some Fuzzy-Wavelet-Like Operators and Their Convergence

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
R. Ezzati ◽  
F. Mokhtarnejad ◽  
N. Hassasi
Keyword(s):  
Uniform Convergence ◽  
Real Number ◽  
Modulus Of Continuity ◽  
Scaling Function ◽  
New Method ◽  
Numerical Examples ◽  
Unit Operator ◽  
Function Of Two Variables

Firstly, we define some new fuzzy-wavelet-like operators via a real-valued scaling function to approximate the continuous fuzzy functions of one and two variables. Then, by using the modulus of continuity, we prove their pointwise and uniform convergence with rates to the fuzzy unit operatorI. Using these fuzzy-wavelet-like operators, we present some numerical examples to illustrate the applicability of the proposed method. Also, we give a new method to approximate the integration of continuous fuzzy real-number-valued function of two variables by using the fuzzy-wavelet-like operator.

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