The parallel sum in C*-algebras

Abstract

Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra with unit e and let a:b be the parallel sum of the two positive definite elements a and b of $\mathcal{A}$ defined by $a(a+b{)}^{-1}b$. We show that the parallel sum a:b can be stated by the solution of the following variational problem: $a:b={\displaystyle \underset{x+y=e}{min}(x^{\ast }\mathrm{ax}+y^{\ast }\mathrm{by}).}$This expression leads us to present some properties of the parallel sum in $C^{\ast }$-algebras and prove inequalities involving it.

Keywords:

• $C^{\ast }$-algebra
• parallel sum
• harmonic mean
• positive definite operator
• operator inequality

2010 Mathematics Subject Classifications:

• 46L05
• 47A30
• 47A63
• 47A64
• 47B65

Acknowledgments

The authors are thankful to the referees for the useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).