scholarly journals Hypo-EP Matrices of Adjointable Operators on Hilbert C ∗ -Modules

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Xiaopeng Li ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Necessary And Sufficient Conditions ◽  
Operator Matrices ◽  
Operator Equations ◽  
Necessary And Sufficient ◽  
Special Circumstances ◽  
Generalized Schur Complement ◽  
Modular Operator

This paper introduces and studies hypo-EP matrices of adjointable operators on Hilbert C ∗ -modules, based on the generalized Schur complement. The necessary and sufficient conditions for some modular operator matrices to be hypo-EP are given, and some special circumstances are also analyzed. Furthermore, an application of the EP operator in operator equations is given.

scholarly journals Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaoji Liu ◽  
Hongwei Jin ◽  
Jelena Višnjić
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement ◽  
Quotient Property

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

scholarly journals The Forward Order Law for Least Squareg-Inverse of Multiple Matrix Products

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 277 ◽  
Author(s):  
Zhiping Xiong ◽  
Zhongshan Liu
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Schur Complement ◽  
Necessary And Sufficient Conditions ◽  
Matrix Product ◽  
The Core ◽  
Matrix Products ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement

The generalized inverse has many important applications in the aspects of the theoretic research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the forward order laws for the generalized inverse of the matrix product. In this paper, by using the extremal ranks of the generalized Schur complement, we obtain some necessary and sufficient conditions for the forward order laws A 1 { 1 , 3 } A 2 { 1 , 3 } ⋯ A n { 1 , 3 } ⊆ ( A 1 A 2 ⋯ A n ) { 1 , 3 } and A 1 { 1 , 4 } A 2 { 1 , 4 } ⋯ A n { 1 , 4 } ⊆ ( A 1 A 2 ⋯ A n ) { 1 , 4 } .

scholarly journals The forward order laws for {1,2,3}- and {1,2,4}-inverses of a three matrix products

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 589-598 ◽  
Author(s):  
Zhongshan Liu ◽  
Zhiping Xiong
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Necessary And Sufficient Conditions ◽  
Matrix Products ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement

In this article, we study the forward order laws for {1,2,3}- and {1,2,4}-inverses of a product of three matrices by using the maximal and minimal ranks of the generalized Schur complement. The necessary and sufficient conditions for A1{1,2,3}A2{1,2,3}A3{1,2,3}? (A1A2A3){1,2,3} and A1{1,2,4}A2 {1,2,4} A3{1,2,4}? (A1A2A3){1,2,4} are presented.

scholarly journals On the solvability of systems of linear equations on commutative semigroup

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Nguyen Thi Thu Huyen ◽  
Nguyen Minh Tuan
Keyword(s):  
Linear Operator ◽  
Linear Space ◽  
Commutative Semigroup ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Systems Of Linear Equations ◽  
Necessary And Sufficient ◽  
Linear Operator Equations

AbstractThis paper deals with the solvability of systems of linear operator equations in a linear space. Namely, the paper provides necessary and sufficient conditions for the operators under which certain kinds of systems of operator equations are solvable.

Generalized left and right Weyl spectra of upper triangular operator matrices

2017 ◽  
Vol 32 ◽  
pp. 41-50
Author(s):  
Guojun Hai ◽  
Dragana Cvetkovic-Ilic
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Operator ◽  
Operator Matrices ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Left And Right ◽  
Necessary And Sufficient

In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$ \bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) $$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a generalized left Weyl operator for each $C\in\B(\K,\H)$, are presented.

scholarly journals Solutions to the System of Operator EquationsA1X=C1,XB2=C2, andA3XB3=C3on HilbertC*-Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaochun Fang ◽  
Enran Hou ◽  
Ge Dong
Keyword(s):  
General Solution ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Existence Of A Solution ◽  
Necessary And Sufficient

We study the solvability of the system of the adjointable operator equationsA1X=C1,XB2=C2, andA3XB3=C3over HilbertC*-modules. We give necessary and sufficient conditions for the existence of a solution and a positive solution of the system. We also derive representations for a general solution and a positive solution to this system. The above results generalize some recent results concerning the equations for operators with closed ranges.

scholarly journals HERIMITIAN SOLUTIONS TO THE EQUATION AXA* + BYB* = C, FOR HILBERT SPACE OPERATORS

2021 ◽  
pp. 001
Author(s):  
Amina Boussaid ◽  
Farida Lombarkia
Keyword(s):  
Operator Equation ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bounded Operators ◽  
Operator Equations ◽  
Common Solution ◽  
Hilbert Space Operators ◽  
Hermitian Solution ◽  
Necessary And Sufficient

Let A, A_{1},  A_{2}, B, B_{1}, B_{2}, C_{1} and C_{2} be linear bounded operators on Hilbert spaces. In this paper, by using generalized inverses, we establish necessary and sufficient conditions for the existence of a common solution and give the form of the general common solution of the operator equations A_{1}XB_{1}=C_{1} and A_{2}XB_{2}=C_{2}, we apply this result to determine new necessary and sufficient conditions for the existence of Hermitian solutions  and give the form of the general Hermitian solution to the operator equation AXB=C. As a consequence, we give necessary and sufficient condition for the existence of Hermitian solution to the operator equation AXA^{*}+BYB^{*}=C.

scholarly journals A NOTE ON SOME SYSTEMS OF GENERALIZED SYLVESTER EQUATIONS

2021 ◽  
pp. 449
Author(s):  
Jovana Nikolov Radenković
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Existence Of A Solution ◽  
Two Systems ◽  
Necessary And Sufficient

In this paper, we study two systems of generalized Sylvester operator equations. We derive necessary and sufficient conditions for the existence of a solution and provide the general form of a solution. We extend some recent results to more general settings.

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