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

Jacek Jachymski. "Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems." Studia Mathematica 195.2 (2009): 99-112. <http://eudml.org/doc/284610>.

@article{JacekJachymski2009,
abstract = {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.},
author = {Jacek Jachymski},
journal = {Studia Mathematica},
keywords = {iterates of linear operators; uniformly stable operator; spectral radius; positive linear operators; univariate Bernstein operators; multivariate Bernstein operators; -Bernstein operators; tensor product Bernstein operators},
language = {eng},
number = {2},
pages = {99-112},
title = {Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems},
url = {http://eudml.org/doc/284610},
volume = {195},
year = {2009},
}

TY - JOUR
AU - Jacek Jachymski
TI - Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems
JO - Studia Mathematica
PY - 2009
VL - 195
IS - 2
SP - 99
EP - 112
AB - 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.
LA - eng
KW - iterates of linear operators; uniformly stable operator; spectral radius; positive linear operators; univariate Bernstein operators; multivariate Bernstein operators; -Bernstein operators; tensor product Bernstein operators
UR - http://eudml.org/doc/284610
ER -