Regular factorizations and perturbation analysis for the core inverse of linear operators in Hilbert spaces

ABSTRACT

This paper concerns the regular factorization and expression of the core inverse in a Hilbert space. Utilizing the regular factorization, we first give some characterizations for the existence and the expression of the group inverse and core inverse. Based on these, we prove that the core inverse of the perturbed operator has the simplest possible expression if and only if the perturbation is range-preserving, and derive an explicit expression under the rank-preserving perturbation. Thus we can conclude that both the range-preserving perturbation and the rank-preserving perturbation are all continuous perturbations. The obtained results extend and improve many recent ones in matrix theory and operator theory.

KEYWORDS:

• Regular factorization
• core inverse
• simplest possible expression
• range-preserving
• rank-preserving
• Hilbert space

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A09
• 47A55
• 65F22

Acknowledgements

The authors would like to express their deep gratitude to the referees for their very detailed and constructive comments and suggestions, which greatly improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (11771378), the Yangzhou University Foundation (2016zqn03) for Young Academic Leaders and the Natural Science Foundation of Yangzhou City (YZ2017123).