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

• Autorzy: Jacek Jachymski
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 47A35: Ergodic theory
• 47B65: Positive operators and order-bounded operators
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 195
• Numer: 2
• Strony: 99-112

Daty

wydano

2009

Twórcy

autor

Jacek Jachymski

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

Identyfikatory

DOI

10.4064/sm195-2-1