Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum
Abstract
Let ℳ be a semifinite von Neumann algebra and T : ℳ → ℳ be a positive L∞−L1 contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on L2(ℳ), is contained in a closed polygon with vertices on the unit circle. In this article, we prove that there exists a positive constant Cp(T) such that labelabstract1stin |supn ≥ 0 + Tn x |p ≤ Cp(T) |x|p for all x ∈ Lp(ℳ), 1<p<∞ extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, similar inequalities as in \eqref{abstract1stin} hold for arbitrary L∞−L1 contractions with the same condition in the numerical range, yielding a vast generalization of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. Moreover, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Bellow \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalization of Stein's method of embedding maximal function into analytic family of operators. However, we show that even in the classical setting, the variational inequality extending \eqref{abstract1stin} holds for arbitrary operators described above, precisely when the spectrum meets the unit circle only at $1.$
Related
- Noncommutative Maximal Ergodic Theorems for Modulated (C,α)-Averages with Operator-Valued Weights
- Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators
- Pointwise Ergodic Theorems Along Reiter Sequences on Locally Compact Groups
- Pointwise Ergodic Theorems Along ⌊ pc⌋ for Every c>1
- Effective Multiplicative Ergodic Theorems for Banach Space Cocycles
BibTeX
@misc{palai2026maximal,
title = {{Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum}},
author = {Subhajit Palai and Samya Kumar Ray},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.16506},
eprint = {2606.16506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16506},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum
AU - Subhajit Palai
AU - Samya Kumar Ray
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.16506
UR - https://arxiv.org/abs/2606.16506
AB - Let ℳ be a semifinite von Neumann algebra and T : ℳ → ℳ be a positive L_∞−L_1 contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on L_2(ℳ), is contained in a closed polygon with vertices on the unit circle. In this article, we prove that there exists a positive constant C_p(T) such that labelabstract1stin |sup_n ≥ 0 ^+ T^n x |_p ≤ C_p(T) |x|_p for all x ∈ L_p(ℳ), 1<p<∞ extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, similar inequalities as in \eqref{abstract1stin} hold for arbitrary L_∞−L_1 contractions with the same condition in the numerical range, yielding a vast generalization of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. Moreover, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Bellow \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalization of Stein's method of embedding maximal function into analytic family of operators. However, we show that even in the classical setting, the variational inequality extending \eqref{abstract1stin} holds for arbitrary operators described above, precisely when the spectrum meets the unit circle only at $1.$
ER -