Relationship between the Hyers-Ulam stability and the Moore-Penrose inverse

In this paper, we establish a link between the Hyers-Ulam stability and the Moore--Penrose inverse, 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 we obtain some characterizations for the Hyers-Ulam stability constants to be continuous. As an application, we give a characterization for the Hyers-Ulam stability constants of the semi-Fredholm operators to be continuous.