A continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse. Issue 19 (16th December 2022) - Chen, Saijie
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 θ, the perturbed operator T ¯ = T + δ T is not necessary θ -invertible and even if it is θ -invertible, lim δ T → 0 T ¯ θ = T θ may not be true. In this paper, we prove that T + T T θ δ T T θ T is θ -invertible and its θ -inverse ( T + T T θ δ T T θ T ) θ has the simplest possible expression, which satisfies lim δ T → 0 ( T + T T θ δ T T θ T ) θ = T θ . Thus, we have found a continuous orbit for the generalized inverse, Moore–Penrose inverse and group inverse.