Result.Math. 53 (2009), 229–236 c  2009 Birkh¨auser Verlag Basel/Switzerland 1422-6383/030229-8, published online June 29, 2009 DOI 10.1007/s00025-008-0333-1 Results in Mathematics Bernstein Polynomials and Operator Theory Catalin Badea Abstract. Kelisky and Rivlin have proved that the iterates of the Bernstein operator (of fixed order) converge to L, the operator of linear interpolation at the endpoints of the interval [0, 1]. In this paper we provide a large class of (not necessarily positive) linear bounded operators T on C[0, 1] for which the iterates T n converge towards L in the operator norm. The proof uses methods from the spectral theory of linear operators. Mathematics Subject Classification (2000). 47A10; 41A35. Keywords. Bounded linear operators, convergence of iterates, spectral theory, Bernstein polynomials. 1. Introduction 1.1. Preamble The Bernstein operator of order m associates to every continuous (real or complex- valued) function f on [0, 1] the m-th Bernstein polynomial B m f (x)= m  k=0 f  k m  b m,k (x) := m  k=0 f  k m  m k  x k (1 − x) m−k . These polynomials were introduced in 1912 in Bernstein’s constructive proof of the Weierstrass approximation theorem. Since then they have been the object of multiple investigations, serving many times as a guide for several theorems in Approximation Theory. The Korovkin theorem [1] is a typical example. In this note, starting from several convergence results for the iterates of B m , we prove the convergence towards L of the iterates of operators from a large class of continuous linear operators acting on C[0, 1]. Here L is the operator of linear interpolation at the endpoints of the interval [0, 1]. The proof uses spectral theory methods. Supported in part by ANR–Projet Blanc DYNOP.