# Universal images of universal elements

Studia Mathematica (2000)

- Volume: 138, Issue: 3, page 241-250
- ISSN: 0039-3223

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topBernal-González, Luis. "Universal images of universal elements." Studia Mathematica 138.3 (2000): 241-250. <http://eudml.org/doc/216702>.

@article{Bernal2000,

abstract = {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.},

author = {Bernal-González, Luis},

journal = {Studia Mathematica},

keywords = {universal element; almost commutativity; universal image; dense range; dense hypercyclic manifold; point spectrum of the adjoint; analytic function of an operator; real entire function; maximal dimension; univeral images; universal elements; hypercyclic operator; dense invariant subspace of maximal algebraic dimension; hypercyclic vectors; Weierstraß factorization theorem},

language = {eng},

number = {3},

pages = {241-250},

title = {Universal images of universal elements},

url = {http://eudml.org/doc/216702},

volume = {138},

year = {2000},

}

TY - JOUR

AU - Bernal-González, Luis

TI - Universal images of universal elements

JO - Studia Mathematica

PY - 2000

VL - 138

IS - 3

SP - 241

EP - 250

AB - 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.

LA - eng

KW - universal element; almost commutativity; universal image; dense range; dense hypercyclic manifold; point spectrum of the adjoint; analytic function of an operator; real entire function; maximal dimension; univeral images; universal elements; hypercyclic operator; dense invariant subspace of maximal algebraic dimension; hypercyclic vectors; Weierstraß factorization theorem

UR - http://eudml.org/doc/216702

ER -

## References

top- [An] S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384-390. Zbl0898.47019
- [Be] L. Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 1003-1010. Zbl0911.47020
- [Bs] J. Bes, Invariant manifolds of hypercyclic vectors for the real scalar case, ibid., 1801-1804. Zbl0914.47005
- [BP] J. Bonet and A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), 587-596. Zbl0926.47011
- [Bo] P. S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), 845-847. Zbl0809.47005
- [Do] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, London, 1978. Zbl0384.47001
- [GS] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. Zbl0732.47016
- [Gr] K. G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987).
- [He] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179-190. Zbl0758.47016
- [HL] G. Herzog und R. Lemmert, Über Endomorphismen mit dichten Bahnen, Math. Z. 213 (1993), 473-477.
- [HS] G. Herzog and C. Schmoeger, On operators T such that ⨍(T) is hypercyclic, Studia Math. 108 (1994), 209-216. Zbl0818.47011
- [Ki] C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto, 1982.
- [Re] C. J. Read, The invariant subspace problem for a class of Banach spaces, 2: hypercyclic operators, Israel J. Math. 63 (1988), 1-40. Zbl0782.47002
- [Ro] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. Zbl0174.44203
- [Ru] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991.
- [Sc] C. Schmoeger, On the norm-closure of the class of hypercyclic operators, Ann. Polon. Math. 65 (1997), 157-161. Zbl0896.47013