On quasi-compact Markov nets

Bartoszek, Wojciech; Erkursun, Nazife

doi:10.1017/s0143385710000350

Cited by 7 publications

(3 citation statements)

References 15 publications

(26 reference statements)

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“…Also for the conditional expectation C n , (I − C n ) n∈N forms an LR-sequence. For more details, we refer to [4] - [7].…”

Section: Lr-sequencesmentioning

confidence: 99%

Asymptotic behavior of operator sequences on KB-spaces

Özcan

2017

Positivity

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The concept of a constrictor was used by several mathematicians to characterize the asymptotic behavior of operators. In this paper we show that a positive LR-sequence on KB-spaces is mean ergodic if the LR-sequence has a weakly compact attractor. Moreover if the weakly compact attractor is an order interval, then a Markovian LR-sequence converges strongly to the finite dimensional fixed space. As a consequence we investigate also stability of LR-sequences of positive operators and existence of lower bound functions on KB-spaces.

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“…Also for the conditional expectation C n , (I − C n ) n∈N forms an LR-sequence. For more details, we refer to [4] - [7].…”

Section: Lr-sequencesmentioning

confidence: 99%

Asymptotic behavior of operator sequences on KB-spaces

Özcan

2017

Positivity

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“…It is worth to remark that the adjoint net to an L R-net is not necessarily an L R-net. Note that in the recent paper, Bartoszek and Erkursun [3], the concept of uniform Lotz-Räbiger nets is investigated.…”

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confidence: 99%

“…There are examples of L R-nets, like a net of local averaging operators (see [9, Example 3.9], [7, Example 5]), which are not related to T -ergodic nets. We refer for mentioned above and many others examples of L R-nets to[3,7,9,10,[16][17][18].…”

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confidence: 99%

Lotz–Räbiger's nets of Markov operators in L1-spaces

Emelyanov

Erkursun

2010

Journal of Mathematical Analysis and Applications

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The Lotz-Räbiger nets (L R-nets) introduced in Räbiger (1993) [18] under the name M-nets provide an appropriate setting for investigation various mean ergodic theorems in Banach spaces. In the present paper we prove several convergence theorems for L R-nets of Markov operators in L 1 -spaces which extend Theorems 1 and 5 from Emel'yanov (2004) [8], and Theorem 1.1 from Lasota (1983) [11]. Preliminaries1.1. Let X be a Banach space, let L(X) be the algebra of all bounded linear operators in X , and let I = I X be the identity operator in X . A family Ψ = (T υ ) υ∈Υ ⊆ L(X) indexed by a directed set Υ = (Υ, ≺) is called an operator net. The net Ψ is strongly convergent if the norm-limit lim υ→∞ T υ x exists for each x. A vector x ∈ X is called a fixed vector for the net Ψ if T υ x = x for every υ ∈ Υ . We denote by Fix(Ψ ) the set of all fixed vectors of Ψ . It is easy to see that Fix(Ψ ) is a closed subspace of X .1.2. The following important concept had been introduced by H.P. Lotz in [16] under the name U M-sequence. Its generalization to arbitrary nets is due to F. Räbiger [18], who used the term M-nets. We prefer to use slightly modified terminology following to the recent paper [5], and call M-nets by Lotz-Räbiger nets. Definition 1. A net Ψ = (T υ ) υ∈Υ ⊆ L(X) is called a Lotz-Räbiger net if

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