Densely hereditarily hypercyclic sequences and large hypercyclic manifolds

Bernal-González, Luis

doi:10.1090/S0002-9939-99-05185-0

Densely hereditarily hypercyclic sequences and large hypercyclic manifolds
HTML articles powered by AMS MathViewer

by Luis Bernal-González PDF

Proc. Amer. Math. Soc. 127 (1999), 3279-3285 Request permission

Abstract:

We prove in this paper that if $(T_{n})$ is a hereditarily hypercyclic sequence of continuous linear mappings between two topological vector spaces $X$ and $Y$, where $Y$ is metrizable, then there is an infinite-dimensional linear submanifold $M$ of $X$ such that each non-zero vector of $M$ is hypercyclic for $(T_{n})$. If, in addition, $X$ is metrizable and separable and $(T_{n})$ is densely hereditarily hypercyclic, then $M$ can be chosen dense.

References

Shamim I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), no. 2, 374–383. MR 1319961, DOI 10.1006/jfan.1995.1036
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
L. BERNAL-GONZÁLEZ, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc., to appear.
L. BERNAL-GONZÁLEZ, Hypercyclic sequences of differential and antidifferential operators, J. Approx. Theory, to appear.
J. BES, Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc., to appear.
J. BONET and A. PERIS, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal., to appear.
Paul S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), no. 3, 845–847. MR 1148021, DOI 10.1090/S0002-9939-1993-1148021-4
Paul S. Bourdon and Joel H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105. MR 1396955, DOI 10.1090/memo/0596
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, DOI 10.1090/S0002-9939-1987-0884467-4
Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
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
Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179–190. MR 1120920, DOI 10.1016/0022-1236(91)90058-D
Gerd Herzog, On a theorem of Seidel and Walsh, Period. Math. Hungar. 30 (1995), no. 3, 205–210. MR 1334965, DOI 10.1007/BF01876619
Robert G. Bartle, The elements of integration, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0200398
C. KITAI, Invariant closed sets for linear operators, Dissertation, University of Toronto, 1982.
Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390. MR 1469346, DOI 10.1006/jfan.1996.3093
Alfonso Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), no. 3, 419–436. MR 1420585, DOI 10.1307/mmj/1029005536
S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22. MR 241956, DOI 10.4064/sm-32-1-17-22