Parallel addition and parallel subtraction of operators

The parallel sum $A:B$ of two invertible nonnegative operators $A$ and $B$ in a Hilbert space $\mathfrak H$ is the operator $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$. This definition was extended to noninvertible operators by Anderson and Duffin for the case $\dim\mathfrak H<\infty$ and by Fillmore and Williams for the general case.
The investigation of parallel addition is continued in this paper; in particular, associativity is proved.
Criteria are established for solvability of the equation $A:X=S$ with an unknown operator $X$ when $A$ and $S$ are given. In the case of solvability, the existence of a minimal solution $S\div A$, called the parallel difference, is proved.
Parallel subtraction in a finite-dimensional space is considered in the last section.
Bibliography: 11 titles.