Perturbation of closed range operators

Abstract

Let T,A be operators with domains D{script}(T) ⊆D{script}(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 ∈D{script}(T). If A{script} 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. © Tübi̇tak.