On a Certain Operator Related to Blackwell’s Markov Chain

We present an example of a densely defined, linear operator on the $$l^{1}$$ l 1 space with the property that each basis vector of the standard Schauder basis of $$l^{1}$$ l 1 does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell’s chain.

SPECTRAL CONDITIONS FOR UNIFORM P-ERGODICITIES OF MARKOV OPERATORS ON ABSTRACT STATES SPACES

In the present paper, we deal with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P-ergodicity, i.e. $\|T^n-P\|\to 0$ , here P is a projection. We have showed that T is uniformly P-ergodic if and only if $\|T^n-P\|\leq C\beta^n$ , $0<\beta<1$ . In this paper, we prove that such a β is characterized by the spectral radius of T − P. Moreover, we give Deoblin’s kind of conditions for the uniform P-ergodicity of Markov operators.

Exploding Markov operators

A special class of doubly stochastic (Markov) operators is constructed. In a sense these operators come from measure preserving transformations and inherit some of their properties, namely ergodicity and positivity of entropy, yet they may have no pointwise factors.

Fixed points results in modular vector spaces with applications to quantum operations and Markov operators

Recently, researchers are showing more interest on both modular vector spaces and modular function spaces. Looking at the number of results it is pertinent to say that, exploration in this direction especially in the area of fixed point theory and applications is still ongoing, many good results can still be unveiled. As a contribution from our part, we study some fixed point results in modular vector spaces associated with order relation. As an application, we were able to study the existence of fixed point(s) of both depolarizing quantum operation and Markov operators through modular functions/modular spaces. The awareness on the importance of quantum theory and Economics globally were the sole motivations of the application choices in our work. Our work complement the existing results. In fact, it adds to the number of application areas that modular vector/function spaces covered.