Geodesic
On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges
Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 240-252.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper, we describe the structure of a strongly continuous operator semigroup $T(t)$ (where $T\colon \mathbb{R}_+ \to \operatorname{End}X$ and $X$ is a complex Banach space) for which $\operatorname{Im}{T(t)}$ is a finite-dimensional space for all $t>0$. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a $C_0$-semigroup.
Keywords: operator semigroup, strong continuity, complex Banach space, Banach algebra, spectrum of an operator, bounded linear operator. 
@article{MZM_2012_91_2_a6,
     author = {A. V. Pechkurov},
     title = {On the {Structure} of a {Semigroup} of {Operators} with {Finite-Dimensional} {Ranges}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {240--252},
     publisher = {mathdoc},
     volume = {91},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a6/}
}
TY  - JOUR
AU  - A. V. Pechkurov
TI  - On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges
JO  - Matematičeskie zametki
PY  - 2012
SP  - 240
EP  - 252
VL  - 91
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a6/
LA  - ru
ID  - MZM_2012_91_2_a6
ER  - 
%0 Journal Article
%A A. V. Pechkurov
%T On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges
%J Matematičeskie zametki
%D 2012
%P 240-252
%V 91
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a6/
%G ru
%F MZM_2012_91_2_a6
A. V. Pechkurov. On the Structure of a Semigroup of Operators with Finite-Dimensional Ranges. Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 240-252. http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a6/

[1] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, M., 1962 | MR | Zbl

[2] N. Danford, Dzh. Shvarts, Lineinye operatory. T. 2. Spektralnaya teoriya, IL, M., 1966 | MR | Zbl

[3] K. Iosida, Funktsionalnyi analiz, Mir, M., 1967 | MR | Zbl

[4] U. Rudin, Funktsionalnyi analiz, Mir, M., 1975 | MR | Zbl

[5] S. G. Krein, K. I. Chernyshov, Singulyarno vozmuschennye differentsialnye uravneniya v banakhovom prostranstve, Preprint, In-t matem. SO AN SSSR, Novosibirsk, 1979

[6] A. G. Baskakov, K. I. Chernyshov, “Spektralnyi analiz lineinykh otnoshenii i vyrozhdennye polugruppy operatorov”, Matem. sb., 193:11 (2002), 3–42 | MR | Zbl

[7] A. G. Baskakov, “Lineinye otnosheniya kak generatory polugrupp operatorov”, Matem. zametki, 84:2 (2008), 175–192 | MR | Zbl

[8] G. A. Sviridyuk, “K obschei teorii polugrupp operatorov”, UMN, 49:4 (1994), 47–74 | MR | Zbl

[9] V. K. Ivanov, I. V. Melnikova, A. I. Filinkov, Differentsialno-operatornye uravneniya i nekorrektnye zadachi, Nauka, M., 1995 | MR | Zbl

[10] A. Favini, A. Yagi, Degenerate Differential Equations in Banach Spaces, Monogr. Textbooks Pure Appl. Math., 215, Marcel Dekker, New York, 1999 | MR | Zbl

[11] R. Cross, Multivalued Linear Operators, Monogr. Textbooks Pure Appl. Math., 213, Marcel Dekker, New York, 1998 | MR | Zbl

[12] M. V. Falaleev, “Fundamentalnye operator-funktsii singulyarnykh differentsialnykh operatorov v banakhovykh prostranstvakh”, Sib. matem. zhurn., 41:5 (2000), 1167–1182 | MR | Zbl

[13] S. P. Zubova, K. I. Chernyshov, “O lineinom differentsialnom uravnenii s fredgolmovym operatorom pri proizvodnoi”, Differentsialnye uravneniya i ikh primenenie, 14, Vilnyus, 1976, 21–39 | MR | Zbl

[14] A. G. Baskakov, Garmonicheskii analiz lineinykh operatorov, Izd. Voronezh. un-ta, Voronezh, 1987 | MR | Zbl

[15] N. Burbaki, Spektralnaya teoriya, Elementy matematiki, Mir, M., 1972 | MR | Zbl

[16] I. V. Kurbatova, “Nekotorye svoistva lineinykh otnoshenii”, Vestn. fakulteta PMM, 2009, no. 7, 68–89

[17] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., 194, Springer-Verlag, New York, 2000 | MR | Zbl

[18] M. Haase, The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl., 169, Birkhäuser Verlag, Basel, 2006 | MR | Zbl