Positive solutions to the equations \(AX=C\) and \(XB=D\) for Hilbert space operators. (English) Zbl 1120.47009

C. G. Khatri and S. K. 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 \(AXB=C\). S. V. Phadke and N. K. 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 \( AA^-A=A\).)


47A62 Equations involving linear operators, with operator unknowns
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
15A24 Matrix equations and identities
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