Acta mathematica scientia, Series B >
HYPERCYCLIC OPERATORS ON QUASI-MAZUR SPACES
Received date: 2007-11-01
Online published: 2010-09-20
Supported by
Supported by the National Natural Science Foundation of China (10571035, 10871141).
Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, K\"{o}the (LF)-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.
QIU Jing-Hui . HYPERCYCLIC OPERATORS ON QUASI-MAZUR SPACES[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1709 -1720 . DOI: 10.1016/S0252-9602(10)60164-6
[1] Ansari S I. Existence of hypercyclic operators on topological vector spaces. J Funct Anal, 1997, 148: 384--390
[2] Bernal-Gonzalez L. On hypercyclic operators in Banach spaces. Proc Amer Math Soc, 1999, 127: 1003--1010
[3] Biersted K D. An introduction to locally convex inductive limits//Hobge-Nlend H, eds. Functional Analysis and Its Applications. Singapore: World Sci, 1988: 35--133
[4] Bonet J, Peris A. Hypercyclic operators on non-normable Fr\'{e}chet spaces. J Funct Anal, 1998, 159: 587--595
[5] Gethner R M, Shapiro J H. Universal vectors for operators on spaces of holomorphic functions. Proc Amer Math Soc, 1987, 100: 281--288
[6] Godefroy G, Shapiro J H. Operators with dense, invariant, cyclic vector manifolds. J Funct Anal, 1991, 98: 229--269
[7] Grosse Erdmann K G. Universal families and hypercyclic operators. Bull Amer Math Soc, 1999, 36: 345--381
[8] Grosse Erdmann K G. Recent developments in hypercyclicity. Rev R Acad Cien Serie A Mat, 2003, 97: 273--286
[9] Horváth J. Topological Vector Spaces and Distributions, Vol 1. Reading, MA: Addison-Wesley, 1966
[10] K\"{o}the G. Topological Vector Spaces I. Berlin: Springer-Verlag, 1969
[11] Kucera J, Mckennon K. Dieudonn\'{e}-Schwartz theorem on bounded sets in inductive limits. Proc Amer Math Soc, 1980, 78: 366--368
[12] Li Ronglu, Cui Chengri, Chao Minhyung. Invariants on all admissible polar topologies. Chin Ann Math, 1998, 19A: 289--294 (in Chinese)
[13] Liu Peide. Foundations of Linear Topological Spaces. Wuhan: Wuhan University Press, 2002 (in Chinese)
[14] Pérez Carreras P, Bonet J. Barrelled Locally Convex Spaces. North-Holland Math Stud, Vol 131. Amsterdam: North-Holland, 1987
[15] Qiu Jinghui. Dieudonné-Schwartz theorem in inductive limits of metrizable spaces. Proc Amer Math Soc, 1984, 92: 255--257
[16] Qiu Jinghui. A general version of Kalton's closed graph theorem. Acta Math Sci, 1995, 15: 161--170
[17] Qiu Jinghui. Local completeness and dual local quasi-completeness. Proc Amer Math Soc, 2000, 129: 1419--1425
[18] Qiu Jinghui. Infra-Mackey spaces, weak barrelledness and barrelledness. J Math Anal Appl, 2004, 292: 459--469
[19] Rolewicz S. On orbits of elements. Studia Math, 1969, 32: 17--22
[20] Salas H. Hypercyclic weighted shifts. Trans Amer Math Soc, 1995, 347: 993--1004
[21] Saxon S A, Sánchez Ruiz L M. Dual local completeness. Proc Amer Math Soc, 1997, 125: 1063--1070
\REF{
[22]} Vogt D. Regularity properties of (LF)-spaces//Bierstedt K D, et al, eds.
Progress in Functional Analysis. North-Holland Math Stud, Vol 170. Amsterdam: North-Holland, 1992: 57--84
\REF{
[23]} Wengenroth J. Derived Functors in Functional Analysis. Berlin, Heidelberg: Springer-Verlag, 2003
\REF{
[24]} Wilansky A. Modern Methods in Topological Vector
Spaces. New York: McGraw-Hill, 1978
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