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

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 Tθ, the perturbed operator T¯=T+δT is not necessary θ-invertible and even if it is θ-invertible, limδT→0T¯θ=Tθ may not be true. In this paper, we prove that T+TTθδTTθT is θ-invertible and its θ-inverse (T+TTθδTTθT)θ has the simplest possible expression, which satisfies limδT→0(T+TTθδTTθT)θ=Tθ. Thus, we have found a continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse.