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Entropy of a doubly stochastic Markov operator and of its shift on the space of trajectories

Paulina Frej — 2012

Colloquium Mathematicae

We define the space of trajectories of a doubly stochastic operator on L¹(X,μ) as a shift space ( X ℕ , ν , σ ) , where ν is a probability measure defined as in the Ionescu-Tulcea theorem and σ is the shift transformation. We study connections between the entropy of a doubly stochastic operator and the entropy of the shift on the space of trajectories of this operator.

An integral formula for entropy of doubly stochastic operators

Bartosz Frej;  Paulina Frej — 2011

Fundamenta Mathematicae

A new formula for entropy of doubly stochastic operators is presented. It is also checked that this formula fulfills the axioms of the axiomatic definition of operator entropy, introduced in an earlier paper of Downarowicz and Frej. As an application of the formula the 'product rule' is obtained, i.e. it is shown that the entropy of a product is the sum of the entropies of the factors. Finally, the proof of continuity of the new 'static' entropy as a function of the measure is given.

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