Linear operators with wild dynamics
Author:
JeanMatthieu Augé
Journal:
Proc. Amer. Math. Soc. 140 (2012), 21032116
MSC (2010):
Primary 47A05; Secondary 47A15, 47A16
Published electronically:
October 20, 2011
MathSciNet review:
2888197
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: If is a separable infinitedimensional Banach space, we construct a bounded and linear operator on such that is not dense and has a nonempty interior with the additional property that can be written , where is the identity and is a compact operator. This answers two recent questions of Hájek and Smith.
 [AH]
S.A. Argyros and R.G. Haydon, An HI space solving the problem, Preprint. http://arxiv.org/abs/0903.3921
 [BM]
Frédéric
Bayart and Étienne
Matheron, Dynamics of linear operators, Cambridge Tracts in
Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318
(2010m:47001)
 [GM]
W.
T. Gowers and B.
Maurey, The unconditional basic sequence
problem, J. Amer. Math. Soc.
6 (1993), no. 4,
851–874. MR 1201238
(94k:46021), 10.1090/S08940347199312012380
 [HS]
Petr
Hájek and Richard
J. Smith, Operator machines on directed graphs, Integral
Equations Operator Theory 67 (2010), no. 1,
15–31. MR
2629974 (2011f:46016), 10.1007/s000200101766y
 [MV]
V.
Müller and J.
Vršovský, On orbitreflexive operators, J. Lond.
Math. Soc. (2) 79 (2009), no. 2, 497–510. MR 2496526
(2010b:47021), 10.1112/jlms/jdn081
 [OP]
R.
I. Ovsepian and A.
Pełczyński, On the existence of a fundamental total
and bounded biorthogonal sequence in every separable Banach space, and
related constructions of uniformly bounded orthonormal systems in
𝐿², Studia Math. 54 (1975), no. 2,
149–159. MR 0394137
(52 #14942)
 [P]
G.T. Prajitura, The geometry of an orbit. Operator theory live, 145154, Theta Ser. Adv. Math., 12, Theta, Bucharest, 2010.
 [R]
C.
J. Read, The invariant subspace problem for a class of Banach
spaces. II. Hypercyclic operators, Israel J. Math. 63
(1988), no. 1, 1–40. MR 959046
(90b:47013), 10.1007/BF02765019
 [T]
L.
Tzafriri, On Banach spaces with unconditional bases, Israel J.
Math. 17 (1974), 84–93. MR 0348456
(50 #954)
 [AH]
 S.A. Argyros and R.G. Haydon, An HI space solving the problem, Preprint. http://arxiv.org/abs/0903.3921
 [BM]
 F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics (2009). MR 2533318 (2010m:47001)
 [GM]
 W.T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6, 851874 (1993). MR 1201238 (94k:46021)
 [HS]
 P. Hájek and R.J. Smith, Operator machines on directed graphs, Integral Equations Oper. Theory 67, no. 1, 1531 (2010). MR 2629974
 [MV]
 V. Müller and J. Vršovský, On orbitreflexive operators, J. London Math. Soc. 79, 497510 (2009). MR 2496526 (2010b:47021)
 [OP]
 R.I. Ovsepian and A. Pelczynski, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in , Studia Math. 54, 149159 (1975). MR 0394137 (52:14942)
 [P]
 G.T. Prajitura, The geometry of an orbit. Operator theory live, 145154, Theta Ser. Adv. Math., 12, Theta, Bucharest, 2010.
 [R]
 C. Read, The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63, 140 (1988). MR 959046 (90b:47013)
 [T]
 L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17, 8493 (1974). MR 0348456 (50:954)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
47A05,
47A15,
47A16
Retrieve articles in all journals
with MSC (2010):
47A05,
47A15,
47A16
Additional Information
JeanMatthieu Augé
Affiliation:
Department of Mathematics, Université Bordeaux 1, 351, cours de la Libération, F33405 Talence cedex, France
Email:
jeanmatthieu.auge@math.ubordeaux1.fr
DOI:
http://dx.doi.org/10.1090/S000299392011110827
Keywords:
Orbits of operators,
compact operators
Received by editor(s):
November 18, 2010
Received by editor(s) in revised form:
February 11, 2011
Published electronically:
October 20, 2011
Communicated by:
Thomas Schlumprecht
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
