On the parallel addition and subtraction of operators on a Hilbert space
Shuaijie Wang, Xiaoyi Tian, Chunyuan Deng · Linear and Multilinear Algebra · 2020
In this paper, we extend the operations of parallel addition A:B and parallel subtraction A÷B from the cone of bounded nonnegative self-adjoint operators to the linear bounded operators on a Hilbert space. Some conditions for the relations A†:B†=(A+B)†,B=(A:B)÷A,(AC)÷(BC)=(A÷B)C,A÷B=(P(A†−B†)P)†,B=A:(B÷A),(CA):(CB)=C(A:B)to be true are studied and the solution of the equation A:X = B is investigated. Moreover, some relationships between the parallel addition and subtraction of projections are obtained.