Norm convergence of some power series of operators in $L^p$ with applications in ergodic theory

Christophe Cuny. "Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory." Studia Mathematica 200.1 (2010): 1-29. <http://eudml.org/doc/285679>.

@article{ChristopheCuny2010,
abstract = {Let X be a closed subspace of $L^\{p\}(μ)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $sup_\{n∈ ℤ\} ||Uⁿ|| < ∞$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $∑_\{n≥1\} (Uⁿf)/n^\{1-α\}$, 0 ≤ α < 1, in terms of $||f + ⋯ + U^\{n-1\}f||_\{p\}$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie.},
author = {Christophe Cuny},
journal = {Studia Mathematica},
keywords = {spectral decomposition; generalized mean ergodic theorems; spectral integration},
language = {eng},
number = {1},
pages = {1-29},
title = {Norm convergence of some power series of operators in $L^\{p\}$ with applications in ergodic theory},
url = {http://eudml.org/doc/285679},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Christophe Cuny
TI - Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 1
SP - 1
EP - 29
AB - Let X be a closed subspace of $L^{p}(μ)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $sup_{n∈ ℤ} ||Uⁿ|| < ∞$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $∑_{n≥1} (Uⁿf)/n^{1-α}$, 0 ≤ α < 1, in terms of $||f + ⋯ + U^{n-1}f||_{p}$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie.
LA - eng
KW - spectral decomposition; generalized mean ergodic theorems; spectral integration
UR - http://eudml.org/doc/285679
ER -