Real positive solutions of operator equations $ AX = C $ and $ XB = D $

Haiyan Zhang; Yanni Dou; Weiyan Yu; Haiyan Zhang; Yanni Dou; Weiyan Yu

doi:10.3934/math.2023777

AIMS Mathematics

2023, Volume 8, Issue 7: 15214-15231. doi: 10.3934/math.2023777

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Research article

Real positive solutions of operator equations $ AX = C $ and $ XB = D $

1.
School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China
2.
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, China
3.
College of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China

Received: 30 December 2022 Revised: 15 April 2023 Accepted: 19 April 2023 Published: 25 April 2023
MSC : 15A09, 47A05

In this paper, we mainly consider operator equations $ AX = C $ and $ XB = D $ in the framework of Hilbert space. A new representation of the reduced solution of $ AX = C $ is given by a convergent operator sequence. The common solutions and common real positive solutions of the system of two operator equations $ AX = C $ and $ XB = D $ are studied. The detailed representations of these solutions are provided which extend the classical closed range case with a short proof.
- reduced solution,
- real positive solution,
- operator equation
Citation: Haiyan Zhang, Yanni Dou, Weiyan Yu. Real positive solutions of operator equations $ AX = C $ and $ XB = D $[J]. AIMS Mathematics, 2023, 8(7): 15214-15231. doi: 10.3934/math.2023777

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Abstract

In this paper, we mainly consider operator equations $ AX = C $ and $ XB = D $ in the framework of Hilbert space. A new representation of the reduced solution of $ AX = C $ is given by a convergent operator sequence. The common solutions and common real positive solutions of the system of two operator equations $ AX = C $ and $ XB = D $ are studied. The detailed representations of these solutions are provided which extend the classical closed range case with a short proof.

References

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