Perturbation of Closed Range Operators.

Let T,A be operators with domains Ɗ(T) ⊆ Ɗ(A) in a normed space X. The operator A is called T-bounded if ∥Ax∥ ≤ a∥x∥+b∥Tx∥ for some a, b ≥ 0 and all x ϵ Ɗ(T). If A has the Hyers-Ulam stability then under some suitable assumptions we show that both T and S := A+T have the Hyers-Ulam stability. We also discuss the best constant of Hyers-Ulam stability for the operator S . Thus we establish a link between T -bounded operators and Hyers-Ulam stability. [ABSTRACT FROM AUTHOR]