The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators
@article{Read1988TheIS, title={The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators}, author={Charles John Read}, journal={Israel Journal of Mathematics}, year={1988}, volume={63}, pages={1-40} }
We continue here the line of investigation begun in [7], where we showed that on every Banach spaceX=l1⊗W (whereW is separable) there is an operatorT with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vectorx inX, the translatesTr x (r=1, 2, 3,...) are dense inX. This is an interesting result even if stated in a form which…
100 Citations
Construction of operators with prescribed behaviour
- Mathematics
- 2003
Abstract.Let X be an infinite dimensional real or complex separable Banach
space, and let $\{v_{n}, n \geq 1\}$ be a dense set of linearly
independent vectors of X. We prove that there exists a…
POWER BOUNDED OPERATORS AND SUPERCYCLIC VECTORS II
- Mathematics
- 2005
We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial…
Orbits of operators and operator semigroups
- Mathematics
- 2010
Denote byB(X) the set of all bounded linear operators acting on a Banach spaceX . For simplicity we assume that all Banach spaces are complex unless stated explicitly otherwise. However, the notions…
On the third problem of Halmos on Banach spaces
- Mathematics
- 2022
A bstract . Assume that X is a complex separable infinite dimensional Banach space and B ( X ) denotes the Banach algebra of all bounded linear operators from X to itself. In 1970, P.R. Halmos raised…
Supercyclic Subspaces: Spectral Theory and Weighted Shifts
- Mathematics
- 2001
Abstract A vector x in a Banach space B is called hypercyclic for a bounded operator T if the orbit { T n x : n ⩾0} is dense in B . If the scalar multiples of the elements in the orbit are dense,…
Invariant Subspaces and the Exponential Map
- Mathematics
- 2004
Bounded operators with no non-trivial closed invariant subspace have been constructed by P. Enflo [6]. In fact, there exist bounded operators on the space 1 with no non-trivial closed invariant…
ORBITS OF OPERATORS
- Mathematics
- 2004
Let X be a Banach space and T ∈ B(X). By an orbit of T we mean a sequence {Tx : n = 0, 1, . . .} where x ∈ X is a fixed vector. Similarly, a weak orbit of T is a sequence {〈Tx, x∗〉 : n = 0, 1, . . .}…
On Read's type operators on Hilbert spaces
- Mathematics
- 2010
Using Read's construction of operators without nontrivial invariant subspaces/subsets on l 1 or C 0 , we construct examples of operators on a Hilbert space whose set of hypercyclic vectors is "large"…
Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators
- Mathematics
- 2006
Let T be a continuous linear operator acting on a Banach space X. We examine whether certain fundamental results for hypercyclic operators are still valid in the Cesàro hypercyclicity setting. In…
References
SHOWING 1-6 OF 6 REFERENCES
On the invariant subspace problem for Banach spaces
- Mathematics
- 1976
© Séminaire analyse fonctionnelle (dit "Maurey-Schwartz") (École Polytechnique), 1975-1976, tous droits réservés. L’accès aux archives du séminaire d’analyse fonctionnelle implique l’accord avec les…