Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld, and Wojciech Kosek. "Positive L¹ operators associated with nonsingular mappings and an example of E. Hille." Colloquium Mathematicae 98.1 (2003): 63-77. <http://eudml.org/doc/285170>.

@article{IsaacKornfeld2003,
abstract = {E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $||Tⁿ|| ∼ n^\{1/4\}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $lim_\{n→ ∞\} ||Tⁿ||/n^\{1-γ\} = ∞. In the class of positive operators these examples are the best possible in the sense that for every such operator T there exists a γ₀ > 0 such that $lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0$. $A class of numerical sequences αₙ, intimately related to the problem of the growth of norms, is introduced, and it is shown that for every sequence αₙ in this class one can get ||Tⁿ|| ≥ αₙ (n = 1,2,...) for some T. Our examples can be realized in a class of positive L¹ operators associated with piecewise linear mappings of [0,1].},
author = {Isaac Kornfeld, Wojciech Kosek},
journal = {Colloquium Mathematicae},
keywords = {mean ergodic theorem; ergodic operator; growth of norms; piecewise linear mappings},
language = {eng},
number = {1},
pages = {63-77},
title = {Positive L¹ operators associated with nonsingular mappings and an example of E. Hille},
url = {http://eudml.org/doc/285170},
volume = {98},
year = {2003},
}

TY - JOUR
AU - Isaac Kornfeld
AU - Wojciech Kosek
TI - Positive L¹ operators associated with nonsingular mappings and an example of E. Hille
JO - Colloquium Mathematicae
PY - 2003
VL - 98
IS - 1
SP - 63
EP - 77
AB - E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $||Tⁿ|| ∼ n^{1/4}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $lim_{n→ ∞} ||Tⁿ||/n^{1-γ} = ∞. In the class of positive operators these examples are the best possible in the sense that for every such operator T there exists a γ₀ > 0 such that $lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0$. $A class of numerical sequences αₙ, intimately related to the problem of the growth of norms, is introduced, and it is shown that for every sequence αₙ in this class one can get ||Tⁿ|| ≥ αₙ (n = 1,2,...) for some T. Our examples can be realized in a class of positive L¹ operators associated with piecewise linear mappings of [0,1].
LA - eng
KW - mean ergodic theorem; ergodic operator; growth of norms; piecewise linear mappings
UR - http://eudml.org/doc/285170
ER -