On the properties of weakly parallel summable operators

In this paper, we consider the properties of operator parallel sum under weaker assumptions, and extend some important results of parallel sum to the case where the operator range R(A+B) is not necessarily closed. Using Douglas theorem, the necessary conditions are given such that CA and CB are weakly parallel summable, and the weak parallel sum CA:CB=C(A:B) is further obtained. Under the restriction of R(A) and R(B) being closed, the several equivalent forms of A:B are given. Finally, the necessary and sufficient condition for C:(−A) to be the solution of A:X=C is provided, the necessary and sufficient conditions of A:B=(A†+B†)† are obtained and the new proof is given.