79 research outputs found
A new class of frequently hypercyclic operators
We study a hypercyclicity property of linear dynamical systems: a bounded
linear operator T acting on a separable infinite-dimensional Banach space X is
said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0}
is dense in X, and frequently hypercyclic if there exists x in X such that for
any non empty open subset U of X, the set {n>0 ; T^n x \in U} has positive
lower density. We prove that if T is a bounded operator on X which has
"sufficiently many" eigenvectors associated to eigenvalues of modulus 1 in the
sense that these eigenvectors are perfectly spanning, then T is automatically
frequently hypercyclic.Comment: 22 pages. To appear in Indiana Univ. Math.
An example of a minimal action of the free semi-group \F^{+}_{2} on the Hilbert space
The Invariant Subset Problem on the Hilbert space is to know whether there
exists a bounded linear operator on a separable infinite-dimensional
Hilbert space such that the orbit of every non-zero
vector under the action of is dense in . We show that there
exists a bounded linear operator on a complex separable
infinite-dimensional Hilbert space and a unitary operator on , such
that the following property holds true: for every non-zero vector ,
either or has a dense orbit under the action of . As a consequence,
we obtain in particular that there exists a minimal action of the free
semi-group with two generators \F^{+}_{2} on a complex separable
infinite-dimensional Hilbert space .Comment: 10
A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces
We present a general method for constructing operators without non-trivial
invariant closed subsets on a large class of non-reflexive Banach spaces. In
particular, our approach unifies and generalizes several constructions due to
Read of operators without non-trivial invariant subspaces on the spaces
, or , and without non-trivial invariant
subsets on . We also investigate how far our methods can be extended
to the Hilbertian setting, and construct an operator on a quasireflexive dual
Banach space which has no non-trivial -closed invariant subspace.Comment: Minor modification
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