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Asymptotic dynamics of generalized Kantorovich operators

Krzysztof Bartoszek, Wojciech Bartoszek

TL;DR

This work characterizes exactly which continuous functions $f$ on a compact space admit uniform convergence of the iterates of a generalized Kantorovich Markov operator $\widehat{T}_i$. By proving a general convergence theorem (Theorem 2.1) for Markov operators with an open dense invariant domain $X_{\lambda}$ and a strongly ergodic complement, it provides necessary and sufficient conditions tied to invariant measures $\lambda$ and Cesàro means. As an application, it resolves Acu and Rasa's conjecture: for $X=[0,1]$ the iterates of $\widehat{T}_i$ converge uniformly on $[0,1]$ exactly when $\int_0^1 f(u)\,du = f(1)$, with the limit being $\big(\int_0^1 f\,d\lambda\big)\mathbf{1}$. This advances understanding of diffusion-type Markov operators and provides a precise criterion linking boundary values to long-term averaging behavior.

Abstract

We characterize the family of continuous functions $f\in C([0,1])$ such that the iterates $\widehat{T}^{k}_{i} f$ converge uniformly on $[0,1]$, where $\widehat{T}_i$ is a generalized Kantorovich operator. This gives an affirmative answer to the problem raised in 2021 by Acu and Rasa.

Asymptotic dynamics of generalized Kantorovich operators

TL;DR

This work characterizes exactly which continuous functions on a compact space admit uniform convergence of the iterates of a generalized Kantorovich Markov operator . By proving a general convergence theorem (Theorem 2.1) for Markov operators with an open dense invariant domain and a strongly ergodic complement, it provides necessary and sufficient conditions tied to invariant measures and Cesàro means. As an application, it resolves Acu and Rasa's conjecture: for the iterates of converge uniformly on exactly when , with the limit being . This advances understanding of diffusion-type Markov operators and provides a precise criterion linking boundary values to long-term averaging behavior.

Abstract

We characterize the family of continuous functions such that the iterates converge uniformly on , where is a generalized Kantorovich operator. This gives an affirmative answer to the problem raised in 2021 by Acu and Rasa.
Paper Structure (3 sections, 2 theorems, 58 equations)

This paper contains 3 sections, 2 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Result
  3. Affirmative answer to Acu and Rasa conjecture

Key Result

Theorem 2.1

Let $X$ be a compact Hausdorff space and $T : C(X) \mapsto C(X)$ be a Markov linear operator satisfying: Then, for a fixed function $f \in C(X)$ the iterates $T^mf$ converge uniformly on $X$ if and only if for all $x\in X\setminus X_{\lambda }$.

Theorems & Definitions (10)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 2.1
  • Theorem 3.1