An invariance property of operator products related to the mixed-type reverse order laws

Abstract

In this paper, we derive an interesting result that the mixed-type reverse order laws hold if and only if the bounded linear operator products is invariant, where is taken, for instance, as , , , , as well as . Moreover, the invariance property of is related to the reverse order law , and we point out that such an invariance property occurs if and only if and are left or and are right invertible operators, or is left and is right invertible operator, where , and are nonzero regular operators.

Keywords:

• generalized inverse
• invariance property
• mixed-type reverse order law
• operator product

AMS Subject Classifications:

• 47A05
• 15A09

Acknowledgements

The authors would like to thank the referees for their helpful comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the GuangXi Natural Science Foundation [grant number 2013GXNSFAA019008]; Project supported by the National Natural Science Foundation of China [grant number 11361009]; and Science Research Project 2013 of the China-ASENA Study Center (GuangXi Science Experiment Center) of GuangXi University for Nationalities. Grants HCIC of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund. The Key Project of Education Department of Guangxi [grant number 201202ZD031].