On the parallel sum of positive operators, forms, and functionals (Q343280)

Let \(A\) and \(B\) be two positive linear operators on a complex Hilbert space \(H\). The parallel sum \(A:B\) of these two operators is the positive operator defined by the nonnegative quadratic form \[ \left\langle (A:B)x,x\right\rangle =\inf_{y\in H} \left\{\left\langle A(x-y),x-y\right\rangle +\left\langle By,y\right\rangle \right\}\text{ for all } x\in H. \] In this paper, the author provides a different construction of the parallel sum. Namely, \(A:B=J_{A} {\kern 1pt} P{\kern 1pt} J_{A}^{\, *} \), where \(P\) is an orthogonal projection of an auxiliary Hilbert space \(Y\) and \(J_{A} \) is a bounded linear operator \(J_{A} :Y\to H\). The technique developed by the author makes it possible to characterize the range of the operator \((A-(A:B))^{1/2} \) .