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## Studia Mathematica

2009 | 195 | 2 | 99-112
Tytuł artykułu

### Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems

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EN
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EN
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 $(Tⁿ)_{n∈ ℕ}$ 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 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.
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Tom
Numer
Strony
99-112
Opis fizyczny
Daty
wydano
2009
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autor
• Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
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