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 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 Banach 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-2-1,
author = {Jacek Jachymski},
title = {Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {99-112},
zbl = {1177.47022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-2-1}
}
Jacek Jachymski. Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems. Studia Mathematica, Tome 192 (2009) pp. 99-112. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-2-1/