Universal images of universal elements

Luis Bernal-González

Studia Mathematica (2000)

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

Abstract

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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.

How to cite

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Bernal-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

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  10. [HL] G. Herzog und R. Lemmert, Über Endomorphismen mit dichten Bahnen, Math. Z. 213 (1993), 473-477. 
  11. [HS] G. Herzog and C. Schmoeger, On operators T such that ⨍(T) is hypercyclic, Studia Math. 108 (1994), 209-216. Zbl0818.47011
  12. [Ki] C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto, 1982. 
  13. [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
  14. [Ro] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. Zbl0174.44203
  15. [Ru] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991. 
  16. [Sc] C. Schmoeger, On the norm-closure of the class of hypercyclic operators, Ann. Polon. Math. 65 (1997), 157-161. Zbl0896.47013

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