## Abstract and Applied Analysis

### Solutions to the System of Operator Equations ${A}_{1}X={C}_{1}$, $X{B}_{2}={C}_{2}$, and ${A}_{3}X{B}_{3}={C}_{3}$ on Hilbert ${C}^{\ast}$-Modules

#### Abstract

We study the solvability of the system of the adjointable operator equations ${A}_{1}X={C}_{1}$, $X{B}_{2}={C}_{2}$, and ${A}_{3}X{B}_{3}={C}_{3}$ over Hilbert ${C}^{\mathrm{\ast}}$-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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 826564, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449679

Digital Object Identifier
doi:10.1155/2013/826564

Mathematical Reviews number (MathSciNet)
MR3132566

#### Citation

Fang, Xiaochun; Hou, Enran; Dong, Ge. Solutions to the System of Operator Equations ${A}_{1}X={C}_{1}$ , $X{B}_{2}={C}_{2}$ , and ${A}_{3}X{B}_{3}={C}_{3}$ on Hilbert ${C}^{\ast}$ -Modules. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 826564, 8 pages. doi:10.1155/2013/826564. https://projecteuclid.org/euclid.aaa/1393449679

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