A continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse

Abstract

As is well known, the generalized inverse, Moore–Penrose inverse and group inverse are not continuous, i.e. for θ = {1, 2}, {1, 2, 3, 4} and {1, 2, 5}, a linear bounded operator T has a θ-inverse $T^{\theta }$, the perturbed operator $\overline{T}=T+\delta T$ is not necessary θ-invertible and even if it is θ-invertible, $\underset{\delta T\to 0}{lim}{\overline{T}}^{\theta }=T^{\theta }$ may not be true. In this paper, we prove that $T+T{T}^{\theta }\delta T{T}^{\theta }T$ is θ-invertible and its θ-inverse $(T+T{T}^{\theta }\delta T{T}^{\theta }T{)}^{\theta }$ has the simplest possible expression, which satisfies $\underset{\delta T\to 0}{lim}(T+T{T}^{\theta }\delta T{T}^{\theta }T{)}^{\theta }=T^{\theta }$. Thus, we have found a continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse.

Keywords:

• Continuous orbit
• generalized inverse
• Moore–Penrose inverse
• group inverse
• the simplest possible expression

2010 Mathematics Subject Classifications:

• 47A55
• 15A09

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11771378, 11871064, 11971419).

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China [11771378, 11871064, 11971419].