# Structure of mixing and category of complete mixing for stochastic operators

Annales Polonici Mathematici (1992)

- Volume: 56, Issue: 3, page 233-242
- ISSN: 0066-2216

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topA. Iwanik, and R. Rębowski. "Structure of mixing and category of complete mixing for stochastic operators." Annales Polonici Mathematici 56.3 (1992): 233-242. <http://eudml.org/doc/262386>.

@article{A1992,

abstract = {Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.},

author = {A. Iwanik, R. Rębowski},

journal = {Annales Polonici Mathematici},

keywords = {Markov operator; conservative deterministic factor; sweeping; stochastic operator on a -finite standard measure space; subinvariant measure; Frobenius-Perron operator; invertible measure preserving transformation; mixing transformation; weak compactness; completely mixing},

language = {eng},

number = {3},

pages = {233-242},

title = {Structure of mixing and category of complete mixing for stochastic operators},

url = {http://eudml.org/doc/262386},

volume = {56},

year = {1992},

}

TY - JOUR

AU - A. Iwanik

AU - R. Rębowski

TI - Structure of mixing and category of complete mixing for stochastic operators

JO - Annales Polonici Mathematici

PY - 1992

VL - 56

IS - 3

SP - 233

EP - 242

AB - Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.

LA - eng

KW - Markov operator; conservative deterministic factor; sweeping; stochastic operator on a -finite standard measure space; subinvariant measure; Frobenius-Perron operator; invertible measure preserving transformation; mixing transformation; weak compactness; completely mixing

UR - http://eudml.org/doc/262386

ER -

## References

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- [16] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York 1974. Zbl0296.47023

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