Densely hereditarily hypercyclic sequences

and large hypercyclic manifolds

Author:
Luis Bernal-González

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3279-3285

MSC (1991):
Primary 47B99; Secondary 46A99, 30E10, 32A07

Published electronically:
May 13, 1999

MathSciNet review:
1646318

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove in this paper that if is a hereditarily hypercyclic sequence of continuous linear mappings between two topological vector spaces and , where is metrizable, then there is an infinite-dimensional linear submanifold of such that each non-zero vector of is hypercyclic for . If, in addition, is metrizable and separable and is densely hereditarily hypercyclic, then can be chosen dense.

**[An1]**Shamim I. Ansari,*Hypercyclic and cyclic vectors*, J. Funct. Anal.**128**(1995), no. 2, 374–383. MR**1319961**, 10.1006/jfan.1995.1036**[An2]**Shamim I. Ansari,*Existence of hypercyclic operators on topological vector spaces*, J. Funct. Anal.**148**(1997), no. 2, 384–390. MR**1469346**, 10.1006/jfan.1996.3093**[Be1]**L. BERNAL-GONZÁLEZ,*On hypercyclic operators on Banach spaces*, Proc. Amer. Math. Soc., to appear. CMP**98:03****[Be2]**L. BERNAL-GONZÁLEZ,*Hypercyclic sequences of differential and antidifferential operators*, J. Approx. Theory, to appear.**[Bes]**J. BES,*Invariant manifolds of hypercyclic vectors for the real scalar case*, Proc. Amer. Math. Soc., to appear. CMP**98:05****[BoP]**J. BONET and A. PERIS,*Hypercyclic operators on non-normable Fréchet spaces*, J. Funct. Anal., to appear.**[Bou]**Paul S. Bourdon,*Invariant manifolds of hypercyclic vectors*, Proc. Amer. Math. Soc.**118**(1993), no. 3, 845–847. MR**1148021**, 10.1090/S0002-9939-1993-1148021-4**[BoS]**Paul S. Bourdon and Joel H. Shapiro,*Cyclic phenomena for composition operators*, Mem. Amer. Math. Soc.**125**(1997), no. 596, x+105. MR**1396955**, 10.1090/memo/0596**[GeS]**Robert M. Gethner and Joel H. Shapiro,*Universal vectors for operators on spaces of holomorphic functions*, Proc. Amer. Math. Soc.**100**(1987), no. 2, 281–288. MR**884467**, 10.1090/S0002-9939-1987-0884467-4**[GoS]**Gilles Godefroy and Joel H. Shapiro,*Operators with dense, invariant, cyclic vector manifolds*, J. Funct. Anal.**98**(1991), no. 2, 229–269. MR**1111569**, 10.1016/0022-1236(91)90078-J**[Gro]**Karl-Goswin Große-Erdmann,*Holomorphe Monster und universelle Funktionen*, Mitt. Math. Sem. Giessen**176**(1987), iv+84 (German). Dissertation, University of Trier, Trier, 1987. MR**877464****[Her]**Domingo A. Herrero,*Limits of hypercyclic and supercyclic operators*, J. Funct. Anal.**99**(1991), no. 1, 179–190. MR**1120920**, 10.1016/0022-1236(91)90058-D**[Hez]**Gerd Herzog,*On a theorem of Seidel and Walsh*, Period. Math. Hungar.**30**(1995), no. 3, 205–210. MR**1334965**, 10.1007/BF01876619**[Hor]**Robert G. Bartle,*The elements of integration*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0200398****[Kit]**C. KITAI,*Invariant closed sets for linear operators*, Dissertation, University of Toronto, 1982.**[LeM]**Fernando León-Saavedra and Alfonso Montes-Rodríguez,*Linear structure of hypercyclic vectors*, J. Funct. Anal.**148**(1997), no. 2, 524–545. MR**1469352**, 10.1006/jfan.1996.3084**[Mon]**Alfonso Montes-Rodríguez,*Banach spaces of hypercyclic vectors*, Michigan Math. J.**43**(1996), no. 3, 419–436. MR**1420585**, 10.1307/mmj/1029005536**[Rol]**S. Rolewicz,*On orbits of elements*, Studia Math.**32**(1969), 17–22. MR**0241956**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
47B99,
46A99,
30E10,
32A07

Retrieve articles in all journals with MSC (1991): 47B99, 46A99, 30E10, 32A07

Additional Information

**Luis Bernal-González**

Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain

Email:
lbernal@cica.es

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05185-0

Keywords:
Hypercyclic vector,
linear operator,
densely hereditarily hypercyclic sequence,
infinite-dimensio\-nal manifold,
dense manifold,
metrizable topological vector space,
entire function of subexponential type,
Runge domain,
infinite order linear differential operator

Received by editor(s):
February 2, 1998

Published electronically:
May 13, 1999

Additional Notes:
This research was supported in part by DGES grant #PB96–1348 and the Junta de Andalucía

Communicated by:
David R. Larson

Article copyright:
© Copyright 1999
American Mathematical Society