scholarly journals The Invariance of the Reverse Order Law under Generalized Inverses of the Product of Two Closed Range Bounded Linear Operators on Hilbert Spaces and Characterization of the Property by the Norm Majorization

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Hanifa Zekraoui ◽  
Cenap Ozel
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Reverse Order Law

scholarly journals Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yang Qi ◽  
Liu Xiaoji ◽  
Yu Yaoming
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Reverse Order ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Reverse Order Law ◽  
Equivalent Conditions

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law A B C † = C † B † A † . Moreover, several equivalent statements of ℛ A A ∗ A B C = ℛ A B C and ℛ C ∗ C A B C ∗ = ℛ A B C ∗ are also deducted by the theory of operators.

scholarly journals Reverse order law of Drazin inverse for bounded linear operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4857-4864 ◽  
Author(s):  
Hua Wang ◽  
Junjie Huang
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Reverse Order ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Reverse Order Law

In this paper, the reverse order law of Drazin inverse is investigated under some conditions in a Banach space. Moreover, the Drazin invertibility of sum for two bounded linear operators are also obtained.

scholarly journals A characterization of real and complex Hilbert spaces among all normed spaces

1983 ◽  
Vol 27 (3) ◽  
pp. 339-345
Author(s):  
J. Vukman
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Identity Operator ◽  
Normed Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Dimensional Range

Let X be a real or complex normed space and L(X) the algebra of all bounded linear operators on X. Suppose there exists a *-algebra B(X) ⊂ L(X) which contains the identity operator I and all bounded linear operators with finite-dimensional range. The main result is: if each operator U ∈ B(X) with the property U*U = UU* = I has norm one then X is a Hilbert space.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

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