Positive definite solutions of certain nonlinear matrix equations

Mousavi, Z.; Mirzapour, F.; Moslehian, Mohammad Sal

doi:10.7153/oam-10-08

Cited by 4 publications

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“…The study of operator equations has been developed from matrices to infinite dimensional spaces; for example, arbitrary Hilbert spaces and Hilbert A-modules, by several mathematicians; see [1,4,8,11,12] and references therein. In [8], some necessary and sufficient conditions for the existence of common Hermitian and positive solutions X ∈ L(H) for the equations AX = C and XB = D are proposed and some formulas for the general forms of their common solutions are given.…”

Section: Introduction Letmentioning

confidence: 99%

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^_3=C_3, A_4XA^_4=C_4$ in Hilbert $C^*$-modules

Eskandari¹,

Fang²,

Moslehian

et al. 2018

ELA

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Necessary and sufficient conditions are given for the operator system $A_1X=C_1$, $XA_2=C_2$, $A_3XA^*_3=C_3$, and $A_4XA^*_4=C_4$ to have a common positive solution, where $A_i$'s and $C_i$'s are adjointable operators on Hilbert $C^*$-modules. This corrects a published result by removing some gaps in its proof. Finally, a technical example is given to show that the proposed investigation in the setting of Hilbert $C^*$-modules is different from that of Hilbert spaces.

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Section: Introduction Letmentioning

confidence: 99%

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^_3=C_3, A_4XA^_4=C_4$ in Hilbert $C^*$-modules

Eskandari¹,

Fang²,

Moslehian

et al. 2018

ELA

Self Cite

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“…Recently several operator equations have been extended from matrices to infinite dimensional spaces, i.e., Hilbert spaces and Hilbert C * -modules; see [11] and references therein. Recall that the notion of Hilbert C * -module is a natural generalization of that of Hilbert space arising by replacing the field of scalars C by a C * -algebra.…”

Section: Introductionmentioning

confidence: 99%

Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

Mousavi

Eskandari

Moslehian

et al. 2017

Linear Algebra and its Applications

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Abstract. Let A, B and C be adjointable operators on a Hilbert C * -module E . Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert C * -modules, we present the general solution of the equation AX + Y B = C when the ranges of A, B and C are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert C * -modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for AXA * + BY B * = C. Finally, we deduce that there exist nonzero operators X, Y ≥ 0 and Z such that AXA * + BY B * = CZ, when A, B and C are given subject to some conditions. Introduction and PreliminariesRecently several operator equations have been extended from matrices to infinite dimensional spaces, i.e., Hilbert spaces and Hilbert C * -modules; see [11] and references therein. Recall that the notion of Hilbert C * -module is a natural generalization of that of Hilbert space arising by replacing the field of scalars C by a C * -algebra. Generalized inverses are useful tools for investigation of solutions of operator equations in the setting of Hilbert C * -modules but these inverses need the strong condition of closedness of ranges of considered operators. Fang et al. [6,7] have studied the solvability of operator equations without the closedness condition on ranges of operators by employing a generalization of a known theorem of Douglas [4, Theorem 1] in the framework of Hilbert C * -modules. In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † . They investigated the equations AX = B Inspired by Fang et al., we investigate the solution of equations AX + Y B = C and AXA * +BY B * = C without the condition of closedness of ranges. This paper is organized as follows. First, we recall some basic information about Hilbert C * -modules. In Section 2, we present an example that shows that the conditions CC * ≤ λAA * for some λ > 0 and R(C) ⊆ R(A) are not equivalent in the setting 2010 Mathematics Subject Classification. 15A24, 46L08, 47A05, 47A62.

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The Investigation on Two Kinds of Nonlinear Matrix Equations

Zhang

2018

Bull. Malays. Math. Sci. Soc.

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Positive definite solutions of certain nonlinear matrix equations

Cited by 4 publications

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Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^_3=C_3, A_4XA^_4=C_4$ in Hilbert $C^*$-modules

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^_3=C_3, A_4XA^_4=C_4$ in Hilbert $C^*$-modules

Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

The Investigation on Two Kinds of Nonlinear Matrix Equations

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Positive definite solutions of certain nonlinear matrix equations

Cited by 4 publications

References 0 publications

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^*_3=C_3, A_4XA^*_4=C_4$ in Hilbert $C^*$-modules

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^*_3=C_3, A_4XA^*_4=C_4$ in Hilbert $C^*$-modules

Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

The Investigation on Two Kinds of Nonlinear Matrix Equations

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Product

Resources

About

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^_3=C_3, A_4XA^_4=C_4$ in Hilbert $C^*$-modules

Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^_3=C_3, A_4XA^_4=C_4$ in Hilbert $C^*$-modules