Let X be a Banach space and T ∈ L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T , that is, for the convergence of the sequence (T n ) n∈N of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ∈ N, the restriction of the pth iterate of T to the range of I − T is a Banach contraction. Our proof is elementary: It uses simple facts from linear algebra, and the

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... nach Contraction Principle. As a consequence, we obtain a theorem on the uniform convergence of iterates of some positive linear operators on C(Ω), which generalizes and subsumes many earlier results including, the Kelisky-Rivlin theorem for univariate Bernstein operators, and its extensions for multivariate Bernstein polynomials over simplices. As another application, we also get a new theorem in this setting giving a formula for the limit of iterates of the tensor product Bernstein operators. 2010 Mathematics Subject Classification: Primary 47A35, 47B38; Secondary 47A10, 47B65. Key words and phrases: iterates of linear operators, uniformly stable operator, spectral radius, positive linear operators, univariate and multivariate Bernstein operators, q-Bernstein operators, tensor product Bernstein operators.