On the positivity of semigroups of operators

Roland Lemmert; Peter Volkmann

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 3, page 483-489
  • ISSN: 0010-2628

Abstract

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In a Banach space E , let U ( t ) ( t > 0 ) be a C 0 -semigroup with generating operator A . For a cone K ⊆ E with non-empty interior we show: ( ☆ )     U ( t ) [ K ] ⊆ K ( t > 0 ) holds if and only if A is quasimonotone increasing with respect to K . On the other hand, if A is not continuous, then there exists a regular cone K ⊆ E such that A is quasimonotone increasing, but ( ☆ ) does not hold.

How to cite

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Lemmert, Roland, and Volkmann, Peter. "On the positivity of semigroups of operators." Commentationes Mathematicae Universitatis Carolinae 39.3 (1998): 483-489. <http://eudml.org/doc/248254>.

@article{Lemmert1998,
abstract = {In a Banach space $E$, let $U(t)$$\,(t>0)$ be a $C_0$-semigroup with generating operator $A$. For a cone $K\subseteq E$ with non-empty interior we show: $(\star )$    $U(t)[K]\subseteq K$$\,(t>0)$ holds if and only if $A$ is quasimonotone increasing with respect to $K$. On the other hand, if $A$ is not continuous, then there exists a regular cone $K\subseteq E$ such that $A$ is quasimonotone increasing, but $(\star )$ does not hold.},
author = {Lemmert, Roland, Volkmann, Peter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semigroups of positive operators; quasimonotonicity; semigroups of positive operators; quasimonotonicity; quasimonotone increasing; -semigroup},
language = {eng},
number = {3},
pages = {483-489},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the positivity of semigroups of operators},
url = {http://eudml.org/doc/248254},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Lemmert, Roland
AU - Volkmann, Peter
TI - On the positivity of semigroups of operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 3
SP - 483
EP - 489
AB - In a Banach space $E$, let $U(t)$$\,(t>0)$ be a $C_0$-semigroup with generating operator $A$. For a cone $K\subseteq E$ with non-empty interior we show: $(\star )$    $U(t)[K]\subseteq K$$\,(t>0)$ holds if and only if $A$ is quasimonotone increasing with respect to $K$. On the other hand, if $A$ is not continuous, then there exists a regular cone $K\subseteq E$ such that $A$ is quasimonotone increasing, but $(\star )$ does not hold.
LA - eng
KW - semigroups of positive operators; quasimonotonicity; semigroups of positive operators; quasimonotonicity; quasimonotone increasing; -semigroup
UR - http://eudml.org/doc/248254
ER -

References

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  8. Kreĭn M.G., Propriétés fondamentales des ensembles coniques normaux dans l'espace de Banach, Doklady Akad. Nauk SSSR 28 (1940), 13-17. (1940) Zbl0024.12202MR0004081
  9. Kreĭn S.G., Lineĭnye differencial'nye uravnenija v banahovom prostranstve, Nauka, Moscow, 1963 (English translation 1971). MR0374949
  10. Volkmann P., Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157-164. (1972) Zbl0226.34058MR0308547
  11. Volkmann P., Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume, Math. Ann. 203 (1973), 201-210. (1973) Zbl0251.34039MR0322305
  12. Volkmann P., Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in Banachräumen, Lecture Notes in Math., vol. 415, Springer, Berlin, 1974, pp.439-443. MR0432995

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