Further results on the (b, c)-inverse, the outer inverse AT,S(2) and the Moore–Penrose inverse in the Banach context

ABSTRACT

In this article, the properties of the (b, c)-inverse, the inverse along an element, the outer inverse with prescribed range and null space $A_{\mathcal{T},\mathcal{S}}^{(2)}$ and the Moore–Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and $C^{\ast }$-algebras. The main properties to be considered are the continuity and the differentiability. In addition, the set of all (b, c)-invertible elements will be proved to be open. The relationship between the (b, c)-inverse and the outer inverse $A_{\mathcal{T},\mathcal{S}}^{(2)}$ will be also characterized.

KEYWORDS:

• (b
• c)-inverse
• outer inverse $A_{\mathcal{T}</Keyword><Keyword>\mathcal{S}}^{(2)}$
• Moore–Penrose inverse
• Banach algebra
• Banach space operator

AMS SUBJECT CLASSIFICATIONS:

• 46H05
• 46L05
• 47A05
• 15A09

Acknowledgements

The author wishes to express his indebtedness with the referee, for her/his suggestions have improved the presentation of this article.

Notes

No potential conflict of interest was reported by the author.