Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
strongly continuous semigroups; Riesz operators; polynomially Riesz operators
strongly continuous semigroups; Riesz operators; polynomially Riesz operators
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.
References:
[1] Caradus S.R.: Operators of Riesz type. Pacific J. Math. 18 (1966), 61-71. MR 0199704 | Zbl 0144.38601
[2] Caradus S.R., Pfaffenberger W.E., Yood B.: Calkin Algebras and Algebras of Operators on Banach Spaces. Marcel Dekker, New York, 1974. MR 0415345 | Zbl 0299.46062
[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
[4] Engel K.J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000. MR 1721989 | Zbl 0952.47036
[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.
[6] Gramsch B., Lay D.: Spectral mapping theorems for essential spectra. Math. Ann. 192 (1971), 17-32. MR 0291846 | Zbl 0203.45601
[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. MR 0780302 | Zbl 0572.47031
[8] Istratescu V.I.: Some remarks on a class of semigroups of operators. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 241-243. MR 0333824
[9] Kato T.: Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anal. Math. 6 (1958), 261-322. MR 0107819 | Zbl 0090.09003
[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. MR 1980178 | Zbl 1069.47051
[11] Latrach K., Paoli J.M.: Polynomially compact-like strongly continuous semigroups. Acta Appl. Math. 82 (2004), 87-99. MR 2061478 | Zbl 1066.47039
[12] Lizama C.: Uniform continuity and compactness for resolvent families of operators. Acta Appl. Math. 38 (1995), 131-138. MR 1326629 | Zbl 0842.45009
[13] Lutz D.: Compactness properties of operator cosine functions. C.R. Math. Rep. Acad. Sci. Canada 2 (1980), 277-280. MR 0600561 | Zbl 0448.47021
[14] Phillips R.S.: Spectral theory for semigroups of linear operators. Trans. Amer. Math. Soc. 71 (1951), 393-415. MR 0044737
[15] Schechter M.: On the essential spectrum of an arbitrary operator I. J. Math. Anal. Appl. 13 (1966), 205-215. MR 0188798 | Zbl 0147.12101
[16] Schechter M.: Principles of Functional Analysis. Graduate Studies in Mathematics, Vol. 36, American Mathematical Society, Providence, 2001. MR 1861991 | Zbl 1002.46002
[17] West T.T.: Riesz operators in Banach spaces. Proc. London Math. Soc. 16 (1966), 131-140. MR 0193522 | Zbl 0139.08401
Search
Partner of
