Range inclusion and operator equations
Let \(A\) and \(D\) be positive operators on a complex Hilbert space \(H\). The operator equation \[ \ell DX+X^*A=A \qquad (\ell>0) \] has a unique solution \(X=C_\ell\) satisfying certain conditions, if there exists a positive operator \(B\) such that \[ AB+ (D-B)A\geq 0. \] The range of \(A\) will be denoted by \(R(A)\). It is proved that the following conjecture is false: If \(R(A+\ell B)\supseteq R(A)\) for all \(\ell\geq 0\), then \(AB+BA\geq 0\).
scientific article; zbMATH DE number 902463