On solution of \(AX+{\alpha{}}XB=A\), range inclusion and parallel sum (Q1209506)

The paper contains theorems connecting range inclusions of bounded linear operators on Hilbert space with the solvability of certerin operator equations. A result about parallel sums is used in the proofs. The first theorem is the following: If \(\text{Range}(A+ B)\supseteq\text{Range}(A)\), then the operator equation \(AX+ X^* B=A\) has a unique solution \(D\) on \(\mathcal H\) so that (a) \(\| D\|^ 2=\inf\{\mu: A^ 2\leq \mu(A+ B)^ 2\}\); (b) \(\text{Ker}(A)= \text{Ker}(D)\); and (c) \(\text{Range}(A+ B)^ -\supseteq\text{Range}(D)\). Moreover \(D\) is positive if and only if \(AB= BA\). The further results of the paper follow from this theorem.