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

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• 作者：Z. Mousavi^a ; ^{mousavi.z@abhariau.ac.ir} ; R. Eskandari^b ; ^{eskandarirasoul@yahoo.com} ; M.S. Moslehian^c ; ^{moslehian@um.ac.ir} ; ^{moslehian@member.ams.org} ; F. Mirzapour^d ; ^{f.mirza@znu.ac.ir}
• 关键词：15A24 ; 46L08 ; 47A05 ; 47A62
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：517
• 期：Complete
• 页码：85-98
• 全文大小：344 K
• 卷排序：517

文摘

Let A, B and C   be adjointable operators on a Hilbert C⁎C⁎-module EE. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert C⁎C⁎-modules, we present the general solution of the equation AX+YB=CAX+YB=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⁎C⁎-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for AXA⁎+BYB⁎=CAXA⁎+BYB⁎=C. Finally, we deduce that there exist nonzero operators X,Y≥0X,Y≥0 and Z   such that AXA⁎+BYB⁎=CZAXA⁎+BYB⁎=CZ, when A, B and C are given subject to some conditions.