We furnish several necessary and sufficient conditions for the following property: For a topological space X, a continuous selfmapping S of X and a family τ of continuous selfmappings of X, the image under S of every τ-universal element is also τ-universal. An application in operator theory, where we extend results of Bourdon, Herrero, Bes, Herzog and Lemmert, is given. In particular, it is proved that every hypercyclic operator on a real or complex Banach space has a dense invariant linear manifold with maximal algebraic dimension consisting, apart from zero, of vectors which are hypercyclic.
@article{bwmeta1.element.bwnjournal-article-smv138i3p241bwm,
author = {Luis Bernal-Gonz\'alez},
title = {Universal images of universal elements},
journal = {Studia Mathematica},
volume = {141},
year = {2000},
pages = {241-250},
zbl = {0959.47002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv138i3p241bwm}
}
Bernal-González, Luis. Universal images of universal elements. Studia Mathematica, Tome 141 (2000) pp. 241-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv138i3p241bwm/
[00000] [An] S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384-390. | Zbl 0898.47019
[00001] [Be] L. Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 1003-1010. | Zbl 0911.47020
[00002] [Bs] J. Bes, Invariant manifolds of hypercyclic vectors for the real scalar case, ibid., 1801-1804. | Zbl 0914.47005
[00003] [BP] J. Bonet and A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), 587-596. | Zbl 0926.47011
[00004] [Bo] P. S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), 845-847. | Zbl 0809.47005
[00005] [Do] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, London, 1978. | Zbl 0384.47001
[00006] [GS] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. | Zbl 0732.47016
[00007] [Gr] K. G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987).
[00008] [He] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179-190. | Zbl 0758.47016
[00009] [HL] G. Herzog und R. Lemmert, Über Endomorphismen mit dichten Bahnen, Math. Z. 213 (1993), 473-477.
[00010] [HS] G. Herzog and C. Schmoeger, On operators T such that ⨍(T) is hypercyclic, Studia Math. 108 (1994), 209-216. | Zbl 0818.47011
[00011] [Ki] C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto, 1982.
[00012] [Re] C. J. Read, The invariant subspace problem for a class of Banach spaces, 2: hypercyclic operators, Israel J. Math. 63 (1988), 1-40. | Zbl 0782.47002
[00013] [Ro] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. | Zbl 0174.44203
[00014] [Ru] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991.
[00015] [Sc] C. Schmoeger, On the norm-closure of the class of hypercyclic operators, Ann. Polon. Math. 65 (1997), 157-161. | Zbl 0896.47013