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.