Convergence of iterates of linear operators and the Kelisky–Rivlin type theorems
Tom 195 / 2009
Streszczenie
Let $X$ be a Banach space and $T\in L(X)$, 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 $(T^n)_{n\in\mathbb N}$ of iterates of $T$ in the uniform topology of $L(X)$. In particular, $T$ is uniformly stable iff for some $p\in\mathbb N$, the restriction of the $p$th 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 Banach Contraction Principle. As a consequence, we obtain a theorem on the uniform convergence of iterates of some positive linear operators on $C({\mit\Omega})$, 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.