# Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators

Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators We consider Markov operators L on C[0, 1] such that for a certain $$c \in [0,1)$$ c ∈ [ 0 , 1 ) , $$\Vert (Lf)' \Vert \le c \Vert f' \Vert$$ ‖ ( L f ) ′ ‖ ≤ c ‖ f ′ ‖ for all $$f \in C^1[0,1]$$ f ∈ C 1 [ 0 , 1 ] . It is shown that L has a unique invariant probability measure $$\nu$$ ν , and then $$\nu$$ ν is used in order to characterize the limit of the iterates $$L^m$$ L m of L. When L is a Kantorovich modification of a certain classical operator from approximation theory, the eigenstructure of this operator is used to give a precise description of the limit of $$L^m$$ L m . This way we extend some known results; in particular, we extend the domain of convergence of the dual functionals associated with the classical Bernstein operator, which gives a partial answer to a problem raised in 2000 by Cooper and Waldron (JAT 105:133–165, 2000, Remark after Theorem 4.20). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators

, Volume 21 (3) – Aug 23, 2016
14 pages

/lp/springer_journal/eigenstructure-and-iterates-for-uniquely-ergodic-kantorovich-PUpy11K2wL
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0441-1
Publisher site
See Article on Publisher Site

### Abstract

We consider Markov operators L on C[0, 1] such that for a certain $$c \in [0,1)$$ c ∈ [ 0 , 1 ) , $$\Vert (Lf)' \Vert \le c \Vert f' \Vert$$ ‖ ( L f ) ′ ‖ ≤ c ‖ f ′ ‖ for all $$f \in C^1[0,1]$$ f ∈ C 1 [ 0 , 1 ] . It is shown that L has a unique invariant probability measure $$\nu$$ ν , and then $$\nu$$ ν is used in order to characterize the limit of the iterates $$L^m$$ L m of L. When L is a Kantorovich modification of a certain classical operator from approximation theory, the eigenstructure of this operator is used to give a precise description of the limit of $$L^m$$ L m . This way we extend some known results; in particular, we extend the domain of convergence of the dual functionals associated with the classical Bernstein operator, which gives a partial answer to a problem raised in 2000 by Cooper and Waldron (JAT 105:133–165, 2000, Remark after Theorem 4.20).

### Journal

PositivitySpringer Journals

Published: Aug 23, 2016

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