{
  "id": "1901.04382",
  "version": "v1",
  "published": "2019-01-14T16:27:55.000Z",
  "updated": "2019-01-14T16:27:55.000Z",
  "title": "On the Asymptotic Behaviour of some Positive Semigroups",
  "authors": [
    "Boris M. Makarow",
    "Martin R. Weber"
  ],
  "comment": "Preprint TU Dresden 19 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Similar to the theory of finite Markov chains it is shown that in a Banach space $X$ ordered by a closed cone $K$ with nonempty interior int($K$) a power bounded positive operator $A$ with compact power such that its trajectories for positive vectors eventually flow into int($K$), defines a \"limit distribution\", i.e. its adjoint operator has a unique fixed point in the dual cone. Moreover, the sequence (A^n) converges with respect to the strong operator topology and for each functional $f\\in X'$ the sequence $((A^*)^n(f))$ converges with respect to the weak*-topology (Theorem 5). If a positive bounded $C_0$-semigroup of linear continuous operators $(S_t)_{t\\geq 0}$ on a Banach space contains a compact operator and the trajectories of the non-zero vectors $x\\in K$ have the property from above then, in particular, $(S_t)$ and $(S^*_t)$ converge to their limit operator with repsect to the operator norm, respectively (Theorem 4). For weakly compact Markov operators in the space of real continuous functions on a compact topological space a corresponding result can be derived, that characterizes the long-term behaviour of regular Markov chains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-14T16:27:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B40",
      "47B65",
      "47D06",
      "37A30"
    ],
    "keywords": [
      "asymptotic behaviour",
      "positive semigroups",
      "regular markov chains",
      "strong operator topology",
      "weakly compact markov operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}