Ulam Stability for the Composition of Operators

Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an example of Ulam stable operators on a Banach space, such that their sum is not Ulam stable. Another example is concerned with a C-0-semigroup (T-t)(t >= 0) of operators for which each T-t is Ulam stable. We present an open problem concerning the Ulam stability of the members of the Bernstein C-0-semigroup. Two other possible problems are mentioned at the end of the paper.