On the parallel addition and subtraction of operators on a Hilbert space

Abstract

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 $\begin{array}{rl}{A}^{†}:B^{†}& =(A+B{)}^{†},B=(A:B)÷A,(AC)÷(BC)=(A÷B)C,\\ A÷B& =\left(P\right(A^{†}-B^{†})P{)}^{†},B=A:(B÷A),(CA):(CB)=C(A:B)\end{array}$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.

Keywords:

• Parallel addition
• parallel subtraction
• positive operator
• projection

2000 Mathematics Subject Classifications:

• 15A09
• 47A05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the National Natural Science Foundation of China [grant number 11671261].