Relationship between the hyers–ulam stability and the moore–penrose inverse

Abstract

In this paper, a link between the Hyers–Ulam stability and the Moore–Penrose inverse is established, that is, a closed operator has the Hyers–Ulam stability if and only if it has a bounded Moore–Penrose inverse. Meanwhile, the stability constant can be determined in terms of the Moore–Penrose inverse. Based on this result, some conditions for the perturbed operators having the Hyers–Ulam stability are obtained, and the Hyers–Ulam stability constant is expressed explicitly in the case of closed operators. In the case of the bounded linear operators, some characterizations for the Hyers–Ulam stability constants to be continuous are derived. As an application, a characterization for the Hyers–Ulam stability constants of the semi-Fredholm operators to be continuous is given.