Solvability of the system of operator equations $$AX=C$$, $$XB=D$$ in Hilbert $$C{{}^{*}}$$-modules

Radenković, Jovana Nikolov; Cvetković-Ilić, Dragana S.; Xu, Qinghua

doi:10.1007/s43034-021-00110-3

Cited by 6 publications

(5 citation statements)

References 12 publications

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“…More precisely, suppose that A ∈ L(F ) and R(A * ) is orthogonally complemented in F . From [15,Remark 1.1], the Moore-Penrose inverse A † can be constructed as the unique operator from R(A)⊕N (A * ) onto R(A * ) such that for every x ∈ R(A * ) and y ∈ N (A * ),…”

Section: Resultsmentioning

confidence: 99%

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Eskandari¹,

Fang²,

Moslehian³

et al. 2021

Preprint

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We extend a work of Pedersen and Takesaki by giving some equivalent conditions for the existence of a positive solution of the so-called Pedersen-Takesaki operator equation XHX = K in the setting of Hilbert C * -modules. It is known that the Douglas lemma does not hold in the setting of Hilbert C * -modules in its general form. In fact, if E is a Hilbert C * -module and A, B ∈ L(E ), then the operator inequality BB * ≤ λAA * with λ > 0 does not ensure that the operator equation AX = B has a solution, in general. We show that under a mild orthogonally complemented condition on the range of operators, AX = B has a solution if and only if BB * ≤ λAA * and R(A) ⊇ R(BB * ). Furthermore, we prove that if L(E ) is a W * -algebra, A, B ∈ L(E ), and R(A * ) = E , then BB * ≤ λAA * for some λ > 0 if and only if R(B) ⊆ R(A). Several examples are given to support the new findings.

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Section: Resultsmentioning

confidence: 99%

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Eskandari¹,

Fang²,

Moslehian³

et al. 2021

Preprint

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show abstract

“…Next, we prove that the general positive solution X is of form (9). According to Theorem 1, X has form (4), where…”

Section: Corollary 1 ([5]mentioning

confidence: 93%

“…The common positive solutions of system (2) were discussed under the condition that A, C, BA * are closed-range operators in [2,3,10], and under the assumption that BA * is a closed range operator in [9]. In this section, we consider the existence and the general form of the positive solutions of system (2) without the restriction on the closed range.…”

Section: The Common Positive Solutions Of Ax = B and XC = Dmentioning

confidence: 99%

“…have been widely studied in matrix algebra [1], the operator space on the Hilbert space [2][3][4][5][6], and the adjointable operator space on Hilbert C * -modules [7][8][9][10]. There is a classical result addressing the existence of solutions for Equation (1), which is the famous Douglas range inclusion theorem [4].…”

Section: Introductionmentioning

confidence: 99%

“…But determining the formulas for the common positive solution remains an open question. Recently, the authors of [9] considered the common positive solutions of system (2) for adjointable operators under certain conditions.…”

Section: Introductionmentioning

confidence: 99%

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Positive Solutions of Operator Equations AX = B, XC = D

Zhang,

Dou,

2023

Axioms

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In this paper, using the technique of operator matrix, we consider the positive solution of the system of operator equations AX=B,XC=D in the framework of the Hilbert space; here, the ranges R(A) of A and R(C) of C are not necessarily closed. Firstly, we provide a new necessary and sufficient condition for the existence of positive solutions of AX=B and also provide a representation of positive solutions, which generalize previous conclusions. Furthermore, using the above result, a condition of equivalence for the existence of common positive solutions of AX=B,XC=D is given, as well as the general forms of positive solutions.

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Common Solutions to the Matrix Equations $$AX=B$$ and $$XC=D$$ on a Subspace

Yuan

2023

J Optim Theory Appl

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Solvability of the system of operator equations $$AX=C$$, $$XB=D$$ in Hilbert $$C{{}^{*}}$$-modules

Cited by 6 publications

References 12 publications

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Positive Solutions of Operator Equations AX = B, XC = D

Common Solutions to the Matrix Equations $$AX=B$$ and $$XC=D$$ on a Subspace

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