Dynamics of random products of positive contractions on L 1

Jalilov, Olimjon, Khursanov, Shokhrukh

Ўзбекистон Миллий университети хабарлари · 2026-yil

Annotatsiya

We study a trajectorywise analogue of the classical zero–two phenomenon for productsof positive contractions on L 1 -spaces. Motivated by the strong zero–two law for a single positivecontraction, we ask whether a uniform bound strictly below 2 for the limiting successive differencesalong a fixed realization forces these differences to converge to zero. We show that this is falsein general. More precisely, for every c ∈ (0,2) we construct two rank-one positive contractions onL 1 ([0,1]) and an alternating realization ω such thatkT n+1ωf − T nω fk 1 = cfor every n ≥ 1 and every nonnegative f with kfk 1 = 1. Hence the hypothesis is satisfied for asuitable ε > 0, while the conclusion fails. In fact, the same equality holds in operator norm. Thusa direct pathwise extension of the deterministic zero–two criterion to non-autonomous, and hencerandom, products is impossible without additional assumptions.

Maqola ma’lumotlari
MualliflarJalilov, Olimjon, Khursanov, Shokhrukh
JurnalЎзбекистон Миллий университети хабарлари
NashriyotЎзбекистон Миллий университети
ISSN2181-7324, 3093-8996
Nashr sanasi2026-10-07
Jild2
Son2.3
Betlar109-112
TilIngliz
Indeksga qo‘shilgan2026-10-08

Maqolaning asl sahifasi

Kalit so‘zlar

positive contractions, zero–two law, random products, L 1 -spaces, rank-one operators, counterexample.

Iqtibos keltirish

APAJalilov, Olimjon, & Khursanov, Shokhrukh (2026). Dynamics of random products of positive contractions on L 1. Ўзбекистон Миллий университети хабарлари, 2(2.3), 109-112. https://journals.nuu.uz/index.php/actanuuz/article/view/15252
GOST (OAK)Jalilov, Olimjon, Khursanov, Shokhrukh Dynamics of random products of positive contractions on L 1 // Ўзбекистон Миллий университети хабарлари. – 2026. – T. 2, № 2.3. – P. 109-112.
MLAJalilov, Olimjon, and Khursanov, Shokhrukh. “Dynamics of random products of positive contractions on L 1.” Ўзбекистон Миллий университети хабарлари, 2(2.3), 109-112, 2026. https://journals.nuu.uz/index.php/actanuuz/article/view/15252
ChicagoJalilov, Olimjon, and Khursanov, Shokhrukh. “Dynamics of random products of positive contractions on L 1.” Ўзбекистон Миллий университети хабарлари, 2(2.3), 109-112 (2026). https://journals.nuu.uz/index.php/actanuuz/article/view/15252
HarvardJalilov, Olimjon, and Khursanov, Shokhrukh (2026) ‘Dynamics of random products of positive contractions on L 1’, Ўзбекистон Миллий университети хабарлари, 2(2.3), 109-112. https://journals.nuu.uz/index.php/actanuuz/article/view/15252
BibTeX
@article{jalilovolimjon2026,
  title = {Dynamics of random products of positive contractions on L 1},
  author = {Jalilov, Olimjon and Khursanov, Shokhrukh},
  journal = {Ўзбекистон Миллий университети хабарлари},
  year = {2026},
  volume = {2},
  number = {2.3},
  pages = {109-112},
  url = {https://journals.nuu.uz/index.php/actanuuz/article/view/15252}
}

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