Positive and real-positive solutions to the equation axa*=c in C*-algebras

Abstract

We obtain new representations for the general positive and real-positive solutions of the equation axa*=c in a C*-algebra using the characterization of positivity based on a matrix representation of an element and the generalized Schur complement. Applications to the equation AXA*=C for operators between Hilbert spaces and for finite matrices are given.

Keywords:

• C*-algebra
• Regular element
• Moore-Penrose inverse
• Equation in a C*-algebra
• Positive solution
• Real-positive solution
• Hilbert space
• Operator equation
• Matrix equation

Keywords:

• AMS Subject Classifications: 16B99
• 16W10
• 15A24
• 47A62

Acknowledgement

The authors are indebted to the referee whose careful reading of the article helped to eliminate several mistakes, and to improve the presentation of the article.