an:05168640
Zbl 1120.47009
Daji??, Alegra; Koliha, J. J.
Positive solutions to the equations \(AX=C\) and \(XB=D\) for Hilbert space operators
EN
J. Math. Anal. Appl. 333, No. 2, 567-576 (2007).
00209049
2007
j
47A62 47A05 15A24
Hilbert space; operator equation; positive solution; common positive solution
\textit{C.\,G.\thinspace Khatri} and \textit{S.\,K.\thinspace Mitra} [SIAM J.~Appl.\ Math.\ 31, 579--585 (1976; Zbl 0359.65033)] studied positive and general solutions of the matrix equations \(AX=C\) and \(XB=D\) and \(AXB=C\). \textit{S.\,V.\thinspace Phadke} and \textit{N.\,K.\thinspace Thakare} [Linear Algebra Appl.\ 23, 191--199 (1979; Zbl 0403.47005)] attempted to describe the Hermitian and positive solutions for Hilbert space operators.
In the paper under review, the authors find some conditions for the existence of Hermitian and positive solutions of the equations \(AX=C\) and \(XB=D\), where \(A, B, C, D\) are bounded linear operators between Hilbert spaces. They show that if \(A\) and \(CA^*\) have closed ranges, then \(AX=C\) has a positive solution \(X\) if and only if \(CA^*\geq 0\) and the range of \(C\) is contained in the range of \(CA^*\). In fact, the general positive solution is given by \(X=C^*(CA^*)^-C+(I-A^-A)S(I-A^-A)^*\), where \(S\) is positive and \((CA^*)^-\) and \(A^-\) are arbitrary inner inverses of \(CA^*\) and \(A\), respectively. (Recall that \(A\) has an inner inverse \(A^-\) if \( AA^-A=A\).)
Maryam Amyari (Mashhad)
Zbl 0359.65033; Zbl 0403.47005