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On the existence of $ J$-class operators on Banach spaces

Author: Amir Bahman Nasseri
Journal: Proc. Amer. Math. Soc. 140 (2012), 3549-3555
MSC (2000): Primary 47A16; Secondary 37B99, 54H20
Published electronically: February 24, 2012
MathSciNet review: 2929023
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Abstract: In this paper we answer in the negative the question raised by G. Costakis and A. Manoussos whether there exists a $ J$-class operator on every non-separable Banach space. In particular we show that there exists a non-separable Banach space constructed by S. Argyros, A. Arvanitakis and A. Tolias such that the $ J$-set of every operator on this space has empty interior for each non-zero vector. On the other hand, on non-separable spaces which are reflexive there always exists a $ J$-class operator.

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  • 1. Y. A. Abramovich, C. D. Aliprantis, An Invitation to Operator Theory, Amer. Math. Soc. (Graduate Studies in Mathematics, V. 50), 2002. MR 1921782 (2003h:47072)
  • 2. S. I. Ansari, Existence of Hypercyclic Operators on Topological Vector Spaces, J. Funct. Anal. 148 (1997), no. 2, 384-390. MR 1469346 (98h:47028a)
  • 3. S. Argyros, A. Arvanitakis, A. Tolias, Saturated Extensions, The Attractors Method and Hereditarily James Tree Spaces, Methods in Banach space theory, 1-90, London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, 2006. MR 2326379 (2009d:46021)
  • 4. M. R. Azimi, V. Müller, A note on $ J$-sets of linear operators, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales 105 (2) (2011), 449-453. MR 2826722
  • 5. T. Bermudez, N. J. Kalton, The range of operators on von Neumann algebras, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1447-1455. MR 1879968 (2003a:47078)
  • 6. L. Bernal-Gonzales, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003-1010. MR 1476119 (99f:47010)
  • 7. G. Costakis, A. Manoussos, $ J$-class operators and hypercyclicity, J. Operator Theory, to
  • 8. J. Lindenstrauss, On non-separable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), no. 6, 967-970. MR 0205040 (34:4875)
  • 9. S. Shkarin, On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc. 137 (2009), no. 1, 123-134. MR 2439433 (2009g:47020)

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Additional Information

Amir Bahman Nasseri
Affiliation: Fakultät für Mathematik, Technische Universität Dortmund, D-44221 Dortmund, Germany

Keywords: $J$-class operators, hypercyclicity
Received by editor(s): September 28, 2010
Received by editor(s) in revised form: April 13, 2011, and April 15, 2011
Published electronically: February 24, 2012
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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