TY - JOUR
T1 - Hypercyclic operators on countably dimensional spaces
AU - Schenke, A.
AU - Shkarin, S.
PY - 2013/5/1
Y1 - 2013/5/1
N2 - According to Grivaux, the group GL(X) of invertible linear operators on a separable infinite dimensional Banach space X acts transitively on the set s (X) of countable dense linearly independent subsets of X. As a consequence, each A? s (X) is an orbit of a hypercyclic operator on X. Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm. We show that for a separable infinite dimensional Fréchet space X, GL(X) acts transitively on s (X) if and only if X possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.
AB - According to Grivaux, the group GL(X) of invertible linear operators on a separable infinite dimensional Banach space X acts transitively on the set s (X) of countable dense linearly independent subsets of X. As a consequence, each A? s (X) is an orbit of a hypercyclic operator on X. Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm. We show that for a separable infinite dimensional Fréchet space X, GL(X) acts transitively on s (X) if and only if X possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.
UR - http://www.scopus.com/inward/record.url?partnerID=yv4JPVwI&eid=2-s2.0-84872968788&md5=b4999e627c9e18a77abb6f1b2469549c
U2 - 10.1016/j.jmaa.2012.11.013
DO - 10.1016/j.jmaa.2012.11.013
M3 - Article
AN - SCOPUS:84872968788
SN - 0022-247X
VL - 401
SP - 209
EP - 217
JO - Journal of Mathematical Analysis and its Applications
JF - Journal of Mathematical Analysis and its Applications
IS - 1
ER -