| Title:
|
A characterization of polynomially Riesz strongly continuous semigroups (English) |
| Author:
|
Latrach, Khalid |
| Author:
|
Paoli, J. Martin |
| Author:
|
Taoudi, M. A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
275-289 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we characterize the class of polynomially Riesz strongly continuous semigroups on a Banach space $X$. Our main results assert, in particular, that the generators of such semigroups are either polynomially Riesz (then bounded) or there exist two closed infinite dimensional invariant subspaces $X_0$ and $X_1$ of $X$ with $X=X_0\oplus X_1$ such that the part of the generator in $X_0$ is unbounded with resolvent of Riesz type while its part in $X_1$ is a polynomially Riesz operator. (English) |
| Keyword:
|
strongly continuous semigroups |
| Keyword:
|
Riesz operators |
| Keyword:
|
polynomially Riesz operators |
| MSC:
|
47B06 |
| MSC:
|
47D03 |
| MSC:
|
47D06 |
| idZBL:
|
Zbl 1150.47325 |
| idMR:
|
MR2241532 |
| . |
| Date available:
|
2009-05-05T16:57:21Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119592 |
| . |
| Reference:
|
[1] Caradus S.R.: Operators of Riesz type.Pacific J. Math. 18 (1966), 61-71. Zbl 0144.38601, MR 0199704 |
| Reference:
|
[2] Caradus S.R., Pfaffenberger W.E., Yood B.: Calkin Algebras and Algebras of Operators on Banach Spaces.Marcel Dekker, New York, 1974. Zbl 0299.46062, MR 0415345 |
| Reference:
|
[3] Cuthbert J.R.: On semigroups such that $U(t)-I$ is compact for some $t>0$.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 9-16. MR 0279625 |
| Reference:
|
[4] Engel K.J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations.Springer, New York, 2000. Zbl 0952.47036, MR 1721989 |
| Reference:
|
[5] Gohberg I.C., Markus A., Feldman I.A.: Normally solvable operators and ideals associated with them.Amer. Math. Soc. Trans. Ser. 2 61 (1967), 63-84. |
| Reference:
|
[6] Gramsch B., Lay D.: Spectral mapping theorems for essential spectra.Math. Ann. 192 (1971), 17-32. Zbl 0203.45601, MR 0291846 |
| Reference:
|
[7] Henriquez H.: Cosine operator families such that $C(t)-I$ is compact for all $t>0$.Indian J. Pure Appl. Math. 16 (1985), 143-152. Zbl 0572.47031, MR 0780302 |
| Reference:
|
[8] Istratescu V.I.: Some remarks on a class of semigroups of operators.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 241-243. MR 0333824 |
| Reference:
|
[9] Kato T.: Perturbation theory for nullity, deficiency and other quantities of linear operators.J. Anal. Math. 6 (1958), 261-322. Zbl 0090.09003, MR 0107819 |
| Reference:
|
[10] Latrach K., Dehici A.: Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results.Int. J. Math. Math. Sci. (2003), 22 1421-1431. Zbl 1069.47051, MR 1980178 |
| Reference:
|
[11] Latrach K., Paoli J.M.: Polynomially compact-like strongly continuous semigroups.Acta Appl. Math. 82 (2004), 87-99. Zbl 1066.47039, MR 2061478 |
| Reference:
|
[12] Lizama C.: Uniform continuity and compactness for resolvent families of operators.Acta Appl. Math. 38 (1995), 131-138. Zbl 0842.45009, MR 1326629 |
| Reference:
|
[13] Lutz D.: Compactness properties of operator cosine functions.C.R. Math. Rep. Acad. Sci. Canada 2 (1980), 277-280. Zbl 0448.47021, MR 0600561 |
| Reference:
|
[14] Phillips R.S.: Spectral theory for semigroups of linear operators.Trans. Amer. Math. Soc. 71 (1951), 393-415. MR 0044737 |
| Reference:
|
[15] Schechter M.: On the essential spectrum of an arbitrary operator I.J. Math. Anal. Appl. 13 (1966), 205-215. Zbl 0147.12101, MR 0188798 |
| Reference:
|
[16] Schechter M.: Principles of Functional Analysis.Graduate Studies in Mathematics, Vol. 36, American Mathematical Society, Providence, 2001. Zbl 1002.46002, MR 1861991 |
| Reference:
|
[17] West T.T.: Riesz operators in Banach spaces.Proc. London Math. Soc. 16 (1966), 131-140. Zbl 0139.08401, MR 0193522 |
| . |
