| Title:
|
On the positivity of semigroups of operators (English) |
| Author:
|
Lemmert, Roland |
| Author:
|
Volkmann, Peter |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
39 |
| Issue:
|
3 |
| Year:
|
1998 |
| Pages:
|
483-489 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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: $(\star)$ \quad $U(t)[K]\subseteq K$ $\,(t>0)$ 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. (English) |
| Keyword:
|
semigroups of positive operators |
| Keyword:
|
quasimonotonicity |
| MSC:
|
47B65 |
| MSC:
|
47D06 |
| idZBL:
|
Zbl 0970.47026 |
| idMR:
|
MR1666770 |
| . |
| Date available:
|
2009-01-08T18:45:33Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119026 |
| . |
| Reference:
|
[1] Arendt W.: Generators of positive semigroups and resolvent positive operators.Habilitationsschrift, Univ. Tübingen, 1984. Zbl 0566.47027, MR 0763347 |
| Reference:
|
[2] Arendt W., Grabosch A., Greiner G., Groh U., Lotz H.P., Moustakas U., Nagel R., Neubrander F., Schlotterbeck U.: One-parameter semigroups of positive operators.Lecture Notes in Math., vol 1184, Springer, Berlin, 1986. MR 0839450 |
| Reference:
|
[3] Borwein J.M., Tingley D.W.: On supportless convex sets.Proc. Amer. Math. Soc. 94 (1985), 471-476. Zbl 0605.46012, MR 0787897 |
| Reference:
|
[4] Fonf V.P.: On supportless convex sets in incomplete normed spaces.Proc. Amer. Math. Soc. 120 (1994), 1173-1176. Zbl 0804.46021, MR 1216811 |
| Reference:
|
[5] Herzog G., Lemmert R.: On quasipositive elements in ordered Banach algebras.Studia Math. 129 (1998), 59-65. Zbl 0908.46032, MR 1611855 |
| Reference:
|
[6] Krasnosel'skiĭM.A.: Pravil'nye i vpolne pravil'nye konusy.Doklady Akad. Nauk SSSR 135 (1960), 255-257. MR 0131157 |
| Reference:
|
[7] Krasnosel'skiĭM.A.: Položitel'nye rešenija operatornyh uravneniĭ.Fizmatgiz, Moscow, 1962 (English translation 1964). |
| Reference:
|
[8] Kreĭn M.G.: Propriétés fondamentales des ensembles coniques normaux dans l'espace de Banach.Doklady Akad. Nauk SSSR 28 (1940), 13-17. Zbl 0024.12202, MR 0004081 |
| Reference:
|
[9] Kreĭn S.G.: Lineĭnye differencial'nye uravnenija v banahovom prostranstve.Nauka, Moscow, 1963 (English translation 1971). MR 0374949 |
| Reference:
|
[10] Volkmann P.: Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen.Math. Z. 127 (1972), 157-164. Zbl 0226.34058, MR 0308547 |
| Reference:
|
[11] Volkmann P.: Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume.Math. Ann. 203 (1973), 201-210. Zbl 0251.34039, MR 0322305 |
| Reference:
|
[12] Volkmann P.: Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in Banachräumen.Lecture Notes in Math., vol. 415, Springer, Berlin, 1974, pp.439-443. MR 0432995 |
| . |
