Let
Let
The operator
The operator
In fact, both notions are equivalent in our setting. This is the content of Birkhoff's Transitivity Theorem; see for instance 1.7 of the instructive notes by Shapiro [
The first version of the
The operator
This suggestive name was coined by Grosse-Erdmann who used it in several talks that he gave years ago. The concept itself was introduced by Godefroy and Shapiro, who showed that it is implied by the Hypercyclicity Criterion [
For a long time all known hypercyclic operators were known to satisfy some version of the Hypercyclicity Criterion. A milestone paper by de la Rosa and Read [
The excellent books by Bayart and Matheron [
Let
It is worth noting that while the author of [
The papers by Salas [
The following three definitions were given in [
The second one is included in their Proposition 2.4.
The third one is their Definition 2.5. (If
In the last display
Proposition 2.3 of [
Proposition 2.4 of [
Theorem
Two comments are in order. Theorem 2.7 of [ When
Assume
The authors of [
It is our goal in this paper to study a “strong’’ version of the Blow-up/Collapse property. We show that for the class of
For convenience, in all what follows, the open neighbourhoods of zero will be chosen to be balanced; that is,
The operators
(a) The Disjoint Blow-up/Collapse property, Definition E, results when
(b) If
Proposition 2.4 of [
Let
Let
Set
As indicated in the introduction, it was pointed out in [
If
Let
Thus (
The following proposition has an immediate proof, and it is often used when studying weighted shifts. Let
Let
The following result says that for the class of
Let
By Theorems 2.1 and 2.2 of [
We prove it for
The converse of Proposition
By Theorems 2.1 and 2.2 of [
If
In the following section we give a partial affirmative answer whenever
In the theorem below, it is worth noting that it is not necessary to suppose beforehand that the operators have a dense set of disjoint hypercyclic vectors; this follows from the construction. However, since
Let
We prove the theorem when
The setting up of the proof is as follows. For each
Let
Let
We now proceed to the construction of the vectors
Step
We choose
Since
Let
The open set
Setting
For
For
We now choose a sufficiently small open set
Since
To finish the proof we have to prove that if
The authors of [
Let
Proposition
Among the following questions the most fundamental is the first one. The next three questions might be easier to handle for the class of
First we recall the relevant definitions. A
The operator
A
Are the Disjoint Blow-up/Collapse property and its strong version equivalent?
Given
Given
Assume that
In the last three questions
Which weighted shifts
Which weighted shifts
Which weighted shifts
More unsolved questions on disjoint hypercyclicity can be found in [
The author would like to thank the referees for their very useful comments. I would also like to thank Professor Dorothy Bollman for her invaluable insights.