Convergence of iterates of linear operators and the Kelisky–Rivlin type theorems

Jacek Jachymski¹

¹ Institute of Mathematics Technical University of Łódź Wólczańska 215 93-005 Łódź, Poland

Studia Mathematica, Tome 195 (2009) no. 2, pp. 99-112

Résumé

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.

Détail

DOI : 10.4064/sm195-2-1

Keywords: banach space space bounded linear operators list necessary sufficient conditions uniform stability convergence sequence mathbb iterates uniform topology particular uniformly stable mathbb restriction pth iterate range i t banach contraction proof elementary uses simple facts linear algebra banach contraction principle consequence obtain theorem uniform convergence iterates positive linear operators mit omega which generalizes subsumes many earlier results including kelisky rivlin theorem univariate bernstein operators its extensions multivariate bernstein polynomials simplices another application get theorem setting giving formula limit iterates tensor product bernstein operators