ON THE HEREDITARILY HYPERCYCLIC OPERATORS

Title & Authors
ON THE HEREDITARILY HYPERCYCLIC OPERATORS
Yousefi, Bahman; Farrokhinia, Ali;

Abstract
Let X be a separable Banach space. We give sufficient conditions under which $\small{T:X{\rightarrow}X}$ is hereditarily hypercyclic. Also, we prove that hereditarily hypercyclicity with respect to a special sequence implies the hereditarily hypercyclicity with respect to the entire sequence.
Keywords
hereditarily hpercyclicity;hypercyclicity criterion;
Language
English
Cited by
1.
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References
1.
T. Bermudez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity cri- teria, Bull. Austral. Math. Soc. 70 (2004), no. 1, 45-54

2.
T. Bermudez, A. Bonilla, J. A. Conejero, and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math. 170 (2005), no. 1, 57-75

3.
T. Bermudez and N. J. Kalton, The range of operators on von Neumann algebras, Proc. Amer. Math. Soc. 13 (2002), no. 5, 1447-1455

4.
L. Bernal-Gonzalez and K. -G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math. 157 (2003), no. 1, 17-32

5.
J. Bes, Three problems on hypercyclic operators, PhD thesis, Kent State Univer- sity, 1998

6.
J. Bes, Three problems on hypercyclic operators, PhD thesis, Kent State Univer- sity, 1998

7.
J. Bonet, Hypercyclic and chaotic convolution operators, J. London Math. Soc.(2) 62 (2000), no. 1, 253-262

8.
J. Bonet, F. Martinez-Gimenez, and A. Peris, Universal and chaotic multipliers on spaces of operators, J. Math. Anal. Appl. 297 (2004), no. 2, 599-611

9.
P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44 (1997), no. 2, 345-353

10.
P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125, Amer. Math. Soc. Providence, RI, 1997

11.
P. S. Bourdon and J. H. Shapiro, Hypercyclic operators that commute with the Bergman backward shift, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5293-5316

12.
G. Costakis and M. Sambarino, Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc. 132 (2004), no. 2, 385-389

13.
C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, 1995

14.
N. S. Feldman, Perturbations of hypercyclic vectors, J. Math. Anal. Appl. 273 (2002), no. 1, 67-74

15.
R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 2, 281-288

16.
G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Func. Anal. 98 (1991), no. 2, 229-269

17.
S. Grivaux, Hypercyclic operators, mixing operators, and the bounded steps prob- lem, J. Operator Theory 54 (2005), no. 1, 147-168

18.
K. G. Grosse-Erdmann, Holomorphic Monster und universelle Funktionen, Mitt. Math. Sem. Giessen No. 176 (1987), iv+84 pp

19.
K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), no. 3, 345-381

20.
K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000), no. 1, 47-68

21.
D. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), no. 1, 93-103

22.
C. Kitai, Invariant closed sets for linear operators, Dissertation, Univ. of Toronto, 1982

23.
F. Leon-Saavedra, Notes about the hypercyclicity criterion, Math. Slovaca 53 (2003), no. 3, 313-319

24.
A. Peris, Hypercyclicity criteria and Mittag-Leffler theorem, Bull. Soc. Roy. Sci. Liuege 70 (2001), no. 4-6, 365-371

25.
A. Peris and L. Saldivia, Syndentically hypercyclic operators, Integral Equations Operator Theory 51 (2005), no. 2, 275-281

26.
S. Rolewics, On orbits of elements, Studia Math. 32 (1969), 17-22

27.
H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993-1004

28.
J. H. Shapiro, Notes on the dynamics of linear operators, http://www.math.msu.edu/ shapiro

29.
A. L. Shields, Weighted shift operators and analytic function theory, Math. Sur- veys, Amer. Math. Soc. Providence 13 (1974), 49-128

30.
B. Yousefi, On the space ${\ell}^P({\beta})$ , Rend. Circ. Mat. Palermo (2) 49 (2000), no. 1, 115-120

31.
B. Yousefi,Unicellularity of the multiplication operator on Banach spaces of formal power series, Studia Math. 147 (2001), no. 3, 201-209

32.
B. Yousefi, Bounded analytic structure of the Banach space of formal power series, Rend. Circ. Mat. Palermo (2) 51 (2002), no. 3, 403-410

33.
B. Yousefi, Strictly cyclic algebra of operators acting on Banach spaces $H^P({\beta})$, Czechoslovak Math. J. 54 (129) (2004), no. 1, 261-266

34.
B. Yousefi,Composition operators on weighted Hardy spaces, Kyungpook Math. J. 44 (2004), no. 3, 319-324

35.
B. Yousefi, On the eighteenth question of Allen Shields, Internat. J. Math. 16 (2005), no. 1, 37-42

36.
B. Yousefi and S. Jahedi, Composition operators on Banach spaces of formal power series, Boll. Unione Math. Ital. Sez. B Artic. Ric. Mat. (8) 6 (2003), no. 2, 481-487

37.
B. Yousefi and A. I. Kashkuli, Cyclicity and unicellularity of the differentiation operator on Banach spaces of formal power series, Math. Proc. R. Ir. Acad. 105 A (2005), no. 1, 1-7

38.
B. Yousefi and H. Rezaei, Hypercyclicity on the algebra of Hilbert-Schmidt oper- ators, Results in Mathematics 46 (2004), no. 1-2, 174-180

39.
B. Yousefi and H. Rezaei, Some necessary and sufficient conditions for hypercyclicity criterion, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 2, 209-216