thmTheorem[section]

Manuel Mancini

11institutetext: Dipartimento di Matematica e Informatica
Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
11email<EMAIL_ADDRESS>ORCID: 0000-0003-2142-6193.

# Biderivations of low-dimensional
Leibniz algebras

Manuel Mancini Supported by University of Palermo and by the “National Group
for Algebraic and Geometric Structures, and their Applications” (GNSAGA –
INdAM).

###### Abstract

In this paper we give a complete classification of the Leibniz algebras of
biderivations of right Leibniz algebras of dimension up to three over a field
$\mathbb{F}$, with $\operatorname{char}(\mathbb{F})\neq 2$. We describe the
main properties of such class of Leibniz algebras and we also compute the
biderivations of the four-dimensional _Dieudonné Leibniz algebra_
$\mathfrak{d_{1}}$. Eventually we give an algorithm for finding derivations
and anti-derivations of a Leibniz algebra as pair of matrices with respect to
a fixed basis.

###### keywords:

Leibniz algebra, Lie algebra, Derivation, Biderivation.

## Introduction

Leibniz algebras were first introduced by J.-L. Loday in [1] as a non-
antisymmetric version of Lie algebras, and many results of Lie algebras were
also established in the frame of Leibniz algebras. Earlier, such algebraic
structures had been considered by A. Blokh, who called them D-algebras [2],
for their strict connection with the derivations. Leibniz algebras play a
significant role in different areas of mathematics and physics.

In [3] derivations of low-dimensional Leibniz algebras have been classified
and studied. After two short preliminary sections, in this paper we aim to
study and classify the Leibniz algebras of biderivations of low-dimensional
Leibniz algebras. Independently of its intrinsic interest, biderivations find
concrete applications in representation theory (cf. [4] and [5]), (sub-)
Riemannian geometry and control theory (see [6], and the bibliography
therein), just to give two instances.

The first section is devoted to some background material on Leibniz algebras
which will be useful for the rest of the paper. We address the reader to [7]
and [8] for more details.

In Section 2 we give the definitions of _derivation_ , _anti-derivation_ and
_biderivations_ for a right Leibniz algebra, which have been first introduced
by J.-L. Loday in [1]. We also recall the main properties of the right and
left adjoint maps and, as an example, we show how to compute the Leibniz
algebra of biderivations of the four-dimensional _Dieudonné Leibniz algebra_
$\mathfrak{d_{1}}$ (cf. [9]).

In the last section we classify the Leibniz algebra of biderivations of two
and three-dimensional right Leibniz algebras over a field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$. We use the classification of low-
dimensional complex Leibniz algebras and their derivations (cf. [7] and [3])
and we give an algorithm for finding derivations and anti-derivations of a
Leibniz algebras as pair of matrices with respect to a fixed basis.

##  1 Leibniz algebras

We assume that $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F})\neq 2$. For the general theory we refer to
[7].

######  Definition .

A _right Leibniz algebra_ over $\mathbb{F}$ is a vector space $L$ over
$\mathbb{F}$ endowed of a bilinear map (called $commutator$ or $bracket$)
$\left[-,-\right]\colon L\times L\rightarrow L$ which satisfies the _right
Leibniz identity_

$\left[\left[x,y\right],z\right]=\left[[x,z],y\right]+\left[x,\left[y,z\right]\right],\;\;\forall
x,y,z\in L.$

In the same way we can define a left Leibniz algebra, using the left Leibniz
identity

$\left[x,\left[y,z\right]\right]=\left[\left[x,y\right],z\right]+\left[y,\left[x,z\right]\right],\;\;\forall
x,y,z\in L.$

A Leibniz algebra that is both left and right is called _symmetric Leibniz
algebra_. From now on we assume that $\dim_{\mathbb{F}}L<\infty$.

Clearly every Lie algebra is a Leibniz algebra and every Leibniz algebra with
skew-symmetric commutator is a Lie algebra. Thus it is defined an adjunction
(see [10]) between the category $\textbf{LieAlg}_{\mathbb{F}}$ of the Lie
algebras over $\mathbb{F}$ and the category $\textbf{LeibAlg}_{\mathbb{F}}$ of
the Leibniz algebras over $\mathbb{F}$. The left adjoint of the inclusion
$i\colon\textbf{LieAlg}_{\mathbb{F}}\rightarrow\textbf{LeibAlg}_{\mathbb{F}}$
is the functor
$\pi\colon\textbf{LeibAlg}_{\mathbb{F}}\rightarrow\textbf{LieAlg}_{\mathbb{F}}$
that associates with every Leibniz algebra $L$ the quotient
$L/\operatorname{Leib}(L)$, where
$\operatorname{Leib}(L)=\langle\left[x,x\right]|\,x\in L\rangle$ is called the
_Leibniz kernel_ of $L$. We observe that $\operatorname{Leib}(L)$ is the
smallest bilateral ideal of $L$ such that $L/\operatorname{Leib}(L)$ is a Lie
algebra. Moreover $\operatorname{Leib}(L)$ is an abelian algebra.

We define the left and the right center of a Leibniz algebra

$\operatorname{Z}_{l}(L)=\left\\{x\in
L\,|\,\left[x,L\right]=0\right\\},\;\;\operatorname{Z}_{r}(L)=\left\\{x\in
L\,|\,\left[L,x\right]=0\right\\}$

and we observe that they coincide when $L$ is a Lie algebra. The _center_ of
$L$ is
$\operatorname{Z}(L)=\operatorname{Z}_{l}(L)\cap\operatorname{Z}_{r}(L)$. In
the case of symmetric Leibniz algebras, the left center and the right center
are bilateral ideals, but in general $\operatorname{Z}_{r}(L)$ is an ideal of
the right Leibniz algebra $L$, meanwhile the left center is not even a
subalgebra.

Finally we recall the definitions of nilpotent and solvable Leibniz algebras.

######  Definition .

Let $L$ be a right Leibniz algebra over $\mathbb{F}$ and let

$L^{(0)}=L,\,\,L^{(k+1)}=[L^{(k)},L],\;\;\forall k\geq 0,$

be the _lower central series of $L$_. $L$ is _$n-$ step nilpotent_ if
$L^{(n-1)}\neq 0$ and $L^{(n)}=0$.

######  Definition .

Let $L$ be a right Leibniz algebra over $\mathbb{F}$ and let

$L^{0}=L,\,\,L^{k+1}=[L^{k},L^{k}],\;\;\forall k\geq 0,$

be the _derived series of $L$_. $L$ is _$n-$ step solvable_ if $L^{n-1}\neq 0$
and $L^{n}=0$.

##  2 Derivations, anti-derivations and biderivations

In this section we recall the definitions of _derivation_ , _anti-derivation_
and _biderivation_ for right Leibniz algebras and we show an example of
computation of the biderivations algebra.

The definition of _derivation_ is the same as in the case of Lie algebras.

######  Definition .

Let $L$ be a Leibniz algebra over $\mathbb{F}$. A _derivation_ of $L$ is a
linear map $d\colon L\rightarrow L$ such that

$d(\left[x,y\right])=\left[d(x),y\right]+\left[x,d(y)\right],\;\;\forall
x,y\in L.$

######  Remark .

Fixed a basis $\\{e_{1},\ldots,e_{n}\\}$ of $L$, where $n=\dim_{\mathbb{F}}L$,
a linear map $d\colon L\rightarrow L$ is a derivation if and only if

$d(\left[e_{i},e_{j}\right])=\left[d(e_{i}),e_{j}\right]+\left[e_{i},d(e_{j})\right],\,\,\forall
i,j=1,\ldots,n.$

The right multiplications are particular derivations called _inner
derivations_ and an equivalent way to define a right Leibniz algebra is to
saying that the (right) adjoint map $\operatorname{ad}_{x}=\left[-,x\right]$
is a derivation, for every $x\in L$. Meanwhile, for a right Leibniz algebra,
the left adjoint maps are not derivations in general.
With the usual bracket $\left[d_{1},d_{2}\right]=d_{1}\circ d_{2}-d_{2}\circ
d_{1}$, the set $\operatorname{Der}(L)$ is a Lie algebra and the set
$\operatorname{Inn}(L)$ of all inner derivations of $L$ is an ideal of
$\operatorname{Der}(L)$. Furthermore, $\operatorname{Aut}(L)$ is a Lie group
and the associated Lie algebra is $\operatorname{Der}(L)$.

The definitions of _anti-derivation_ and _biderivation_ for a Leibniz algebra
were first given by J.-L. Loday in [1].

######  Definition .

An _anti-derivation_ of a right Leibniz algebra $L$ is a linear map $D\colon
L\rightarrow L$ such that

$D([x,y])=[D(x),y]-[D(y),x],\;\;\forall x,y\in L.$

For a left Leibniz algebra we have to ask that

$D([x,y])=[x,D(y)]-[y,D(x)],\;\;\forall x,y\in L.$

We observe that in the case of Lie algebras, there is no difference between a
derivation and an anti-derivation. Moreover one can check that, for every
$x\in L$, the left adjoint map $\operatorname{Ad}_{x}=[x,-]$ defines an anti-
derivation.

######  Remark .

The set of anti-derivations of a Leibniz algebra $L$ has a
$\operatorname{Der}(L)-$module structure with the action

$d\cdot D:=[D,d]=D\circ d-d\circ D,$

for every $d\in\operatorname{Der}(L)$ and for every anti-derivation $D$.

######  Remark .

Let $D\colon L\rightarrow L$ be an anti-derivation. Then, for every $x\in L$,
we have

$D([x,x])=[D(x),x]-[D(x),x]=0,$

thus $D(\operatorname{Leib}(L))=0$.

######  Definition .

Let $L$ be a right Leibniz algebra. A _biderivation_ of $L$ is a pair

$(d,D)$

where $d$ is a derivation and $D$ is an anti-derivation, such that

$[x,d(y)]=[x,D(y)],\;\;\forall x,y\in L.$ (1)

The set of all biderivations of $L$, denoted by $\operatorname{Bider}(L)$, has
a Leibniz algebra structure with the bracket

$[(d,D),(d^{\prime},D^{\prime})]=(d\circ d^{\prime}-d^{\prime}\circ d,D\circ
d^{\prime}-d^{\prime}\circ
D),\;\;\forall(d,D),(d,D^{\prime})\in\operatorname{Bider}(L),$

and it is possible to define a Leibniz algebra morphism

$L\rightarrow\operatorname{Bider}(L)$

by

$x\mapsto(-\operatorname{ad}_{x},\operatorname{Ad}_{x}),\;\;\forall x\in L.$

The pair $(-\operatorname{ad}_{x},\operatorname{Ad}_{x})$ is called _inner
biderivation_ of $L$ and the set of all inner biderivations forms a Leibniz
subalgebra of $\operatorname{Bider}(L)$.

Now we give an example of computation of the biderivations of a Leibniz
algebra.

######  Example .

Let $\mathfrak{d}_{1}$ be the four-dimensional _Dieudonné Leibniz algebra_
(see [9] for more details), i.e. $\mathfrak{d}_{1}$ has basis
$\\{e_{1},e_{2},e_{3},z\\}$ and non-zero commutators

$[e_{1},e_{3}]=[e_{2},e_{3}]=-[e_{3},e_{1}]=[e_{3},e_{2}]=z.$

We want to find the Leibniz algebra $\operatorname{Bider}(\mathfrak{d}_{1})$
of biderivations of $\mathfrak{d}_{1}$. The derivations of the Dieudonné
Leibniz algebra $\mathfrak{d_{n}}$ over a field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$, have been found in [11] and it turns
out that

$\operatorname{Der}(\mathfrak{d_{1}})=\left\\{\begin{pmatrix}x&0&\alpha&0\\\
0&x&0&0\\\ 0&0&y&0\\\
a_{1}&a_{2}&a_{3}&x+y\end{pmatrix}\Bigg{|}x,y,\alpha,a_{1},a_{2},a_{3}\in\mathbb{F}\right\\}$

Now let $D\in\operatorname{gl}(\mathfrak{d}_{1})$ be an anti-derivation, then
$D$ is represented, with respect to the basis $\\{e_{1},e_{2},e_{3},z\\}$, by
a matrix of the type

$\begin{pmatrix}a_{11}&a_{12}&a_{13}&0\\\ a_{21}&a_{22}&a_{23}&0\\\
a_{31}&a_{32}&a_{33}&0\\\ A_{1}&A_{2}&A_{3}&0\end{pmatrix}$

where $D(z)=0$ because
$\operatorname{Leib}(\mathfrak{d}_{1})=[\mathfrak{d}_{1},\mathfrak{d}_{1}]=\mathbb{F}z$,
and the entries $a_{ij}$, $i,j=1,2,3$, must satisfy the following equations

$\displaystyle a_{31}+a_{32}=0,$ $\displaystyle a_{11}+a_{21}+a_{33}=0,$
$\displaystyle a_{12}+a_{22}-a_{33}=0,$ $\displaystyle-
a_{12}-a_{22}-a_{33}=0.$

Thus a general anti-derivation of $\mathfrak{d}_{1}$ is represented by

$\begin{pmatrix}a_{11}&a_{12}&a_{13}&0\\\
a_{21}&a_{11}+a_{21}-a_{12}&a_{23}&0\\\ a_{31}&-a_{31}&a_{11}+a_{21}&0\\\
A_{1}&A_{2}&A_{3}&0\end{pmatrix}$

and, by applying the condition $(\ref{1})$, we obtain

$a_{31}=0,\;\alpha=a_{13}-a_{23},\;y=a_{11}+a_{21}$

and

$x=a_{11}-a_{21}=a_{11}+a_{21}-2a_{12}.$

We conclude that

$\operatorname{Bider}(\mathfrak{d}_{1})=\left\\{\left(\begin{pmatrix}x&0&\alpha&0\\\
0&x&0&0\\\ 0&0&y&0\\\
a_{1}&a_{2}&a_{3}&x+y\end{pmatrix},\begin{pmatrix}\frac{y+x}{2}&\frac{y-x}{2}&\alpha+\beta&0\\\
\frac{y-x}{2}&\frac{y+x}{2}&\beta&0\\\ 0&0&y&0\\\
A_{1}&A_{2}&A_{3}&0\end{pmatrix}\right)\Bigg{|}x,y,\alpha,\beta,a_{i},A_{j}\in\mathbb{F}\right\\}$

and the inner biderivations are represented by the pairs of matrices of type

$\left(\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
a_{1}&a_{1}&a_{3}&0\end{pmatrix},\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ a_{1}&-a_{1}&A_{3}&0\end{pmatrix}\right).$

##  3 Biderivations of low-dimensional Leibniz algebras

Now we want to study in detail the Leibniz algebras of biderivations of low-
dimensional non-Lie Leibniz algebras over a field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$. There is no non-trivial Leibniz
algebra in dimension 1, thus we start with two-dimensional Leibniz algebras.

###  3.1 Biderivations of two-dimensional Leibniz algebras

Let $\operatorname{dim}_{\mathbb{F}}L=2$, i.e.
$L=\mathbb{F}e_{1}+\mathbb{F}e_{2}$, then, as shown in [12] by C. Cuvier, up
to isomorphism we have only two non-Lie Leibniz algebra structures on $L$.

  1. 1.

$L_{1}$ : nilpotent Leibniz algebra with non-trivial bracket
$[e_{2},e_{2}]=e_{1}$;

  2. 2.

$L_{2}$ : solvable Leibniz algebra with the table of multiplication
$[e_{1},e_{2}]=[e_{2},e_{2}]=e_{1}$.

Notice that $L_{1}$ is a symmetric Leibniz algebra, meanwhile $L_{2}$ is only
a right Leibniz algebra. It turns out that

$\operatorname{Der}(L_{1})=\left\\{\begin{pmatrix}2a&b\\\ 0&a\\\
\end{pmatrix}\bigg{|}a,b\in\mathbb{F}\right\\}$

and

$\operatorname{Der}(L_{2})=\left\\{\begin{pmatrix}a&a\\\ 0&0\\\
\end{pmatrix}\bigg{|}a\in\mathbb{F}\right\\}.$

Moreover it is easy to check that the set of anti-derivations of $L_{1}$ and
$L_{2}$ are both represented by the matrices of the form

$\begin{pmatrix}0&x\\\ 0&y\\\ \end{pmatrix}.$

The condition $(\ref{1})$ implies that $y=a$ for $L_{1}$ and $y=0$ for
$L_{2}$, thus we have

$\operatorname{Bider}(L_{1})=\left\\{\left(\begin{pmatrix}2a&b\\\ 0&a\\\
\end{pmatrix},\begin{pmatrix}0&x\\\ 0&a\\\
\end{pmatrix}\right)\bigg{|}a,b,x\in\mathbb{F}\right\\}$

and

$\operatorname{Bider}(L_{2})=\left\\{\left(\begin{pmatrix}a&a\\\ 0&0\\\
\end{pmatrix},\begin{pmatrix}0&x\\\ 0&0\\\
\end{pmatrix}\right)\bigg{|}a,x\in\mathbb{F}\right\\}.$

Finally the inner biderivations of $L_{1}$ are represented by the pairs of
matrices

$\left(\begin{pmatrix}0&b\\\ 0&0\\\ \end{pmatrix},\begin{pmatrix}0&-b\\\
0&0\\\ \end{pmatrix}\right),$

meanwhile the biderivations of $L_{2}$ are all inner.

###  3.2 Biderivations of three-dimensional Leibniz algebras

Three-dimensional complex Leibniz algebras and their derivations have been
classified in [13] and [3], meanwhile the more general classification of
three-dimensional right Leibniz algebras over a field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$, can be found in [14] and [15].

Let $\dim_{\mathbb{F}}L=3$ and let $\\{e_{1},e_{2},e_{3}\\}$ be a basis of $L$
over $\mathbb{F}$. The list of non-isomorphic three-dimensional right Leibniz
algebras over $\mathbb{F}$ is the following.

Leibniz algebra | Non-zero brackets
---|---
$L_{1}$ | $[e_{1},e_{3}]=-2e_{1},\;[e_{2},e_{2}]=e_{1},\;[e_{3},e_{2}]=-[e_{2},e_{3}]=e_{2}$
$L_{2}(\alpha),\;\alpha\neq 0$ | $[e_{1},e_{3}]=\alpha e_{1},\;[e_{3},e_{2}]=-[e_{2},e_{3}]=e_{2}$
$L_{3}$ | $[e_{3},e_{2}]=-[e_{2},e_{3}]=e_{2},\;[e_{3},e_{3}]=-e_{1}$
$L_{4}$ | $[e_{2},e_{2}]=e_{1},\;[e_{3},e_{3}]=e_{1}$
$L_{5}$ | $[e_{2},e_{2}]=e_{1},\;[e_{3},e_{3}]=-e_{1}$
$L_{7}(\alpha),\;\alpha\neq 0$ | $[e_{2},e_{2}]=[e_{2},e_{3}]=e_{1},\;[e_{3},e_{3}]=\alpha e_{1}$
$L_{8}$ | $[e_{2},e_{3}]=e_{1}$
$L_{9}$ | $[e_{1},e_{3}]=e_{2},\;[e_{2},e_{3}]=e_{1}$
$L_{10}$ | $[e_{1},e_{3}]=e_{2},\;[e_{2},e_{3}]=-e_{1}$
$L_{12}(\alpha),\;\alpha\neq 0$ | $[e_{1},e_{3}]=e_{2},\;[e_{2},e_{3}]=\alpha e_{1}+e_{2}$
$L_{13}$ | $[e_{1},e_{3}]=e_{1},\;[e_{2},e_{3}]=e_{2}$
$L_{14}$ | $[e_{1},e_{3}]=e_{2},\;[e_{3},e_{3}]=e_{1}$
$L_{15}$ | $[e_{1},e_{3}]=e_{1}+e_{2},\;[e_{3},e_{3}]=e_{1}$

Here we use the same notation of [15], but we do not report the algebras
$L_{6}(\alpha)$ and $L_{11}(\alpha)$, where $\alpha\neq 0$, which are
isomorphic to $L_{4}$ and $L_{9}$ respectively. We want to extend the results
of [3] by completing the classification of the Lie algebras of derivations of
three-dimensional Leibniz algebras over a general field $\mathbb{F}$, with
$\operatorname{char}(\mathbb{F})\neq 2$, and by finding the biderivations of
this class of Leibniz algebras.

######  Remark .

In this section we use the following algorithm for finding derivations and
anti-derivations. Let $L$ be a Leibniz algebra and let
$(d,D)\in\operatorname{Bider}(L)$. Then, for every $x,y\in L$, we have

$d([x,y])=[d(x),y]+[x,d(y)],\;\;D([x,y])=[D(x),y]-[D(y),x]$

if and only if

$(d\circ\operatorname{ad}_{y})(x)=(\operatorname{ad}_{y}\circ
d)(x)+\operatorname{ad}_{d(y)}(x),\;\;(D\circ\operatorname{ad}_{y})(x)=(\operatorname{ad}_{y}\circ
D)(x)-\operatorname{Ad}_{D(y)}(x),$

thus

$[d,\operatorname{ad}_{y}]=\operatorname{ad}_{d(y)},\;\;[D,\operatorname{ad}_{y}]=-\operatorname{Ad}_{D(y)}.$

Fixed a basis $\\{e_{1},\cdots,e_{n}\\}$ of $L$, we have that the biderivation
$(d,D)$ is represented by a pair of $n\times n$ matrices
$((d_{i,j})_{i,j},(D_{i,j})_{i,j})$ and for every $i=1,\ldots,n$

$[d,\operatorname{ad}_{e_{i}}]=\operatorname{ad}_{d(e_{i})},\;\;[D,\operatorname{ad}_{e_{i}}]=-\operatorname{Ad}_{D(e_{i})},$

which are equations in the entries of the matrices representing $d$ and $D$.
By solving this set of equations, and after imposing the compatibility
condition (1), we find the matrices $(d_{i,j})_{i,j},(D_{i,j})_{i,j}$.

A straightforward application of the above algorithm produces the following
complete classification of biderivations of three-dimensional right Leibniz
algebras over $\mathbb{F}$. In particular we find that the dimension of these
biderivation algebras lies between three and six and there are only two
parameterized families of Leibniz algebra of biderivations of three-
dimensional Leibniz algebras over $\mathbb{F}$.

######  Theorem 1.

Let $\mathbb{F}$ be a field with $\operatorname{char}(\mathbb{F})\neq 2$. The
Leibniz algebras of biderivations of three-dimensional right Leibniz algebras
over $\mathbb{F}$ can be described as follows.

  * •

$\operatorname{Bider}(L_{1})=\left\\{\left(\begin{pmatrix}2x&y&0\\\ 0&x&y\\\
0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&-y&a\\\ 0&x&y\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,a\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{2}(\alpha))=\left\\{\left(\begin{pmatrix}x&0&0\\\
0&y&z\\\ 0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&a&b\\\ 0&y&z\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,a,b\in\mathbb{F}\right\\}$,

if $\alpha\neq-1$ and

$\operatorname{Bider}(L_{2}(-1))=\left\\{\left(\begin{pmatrix}x&0&0\\\
0&y&z\\\ 0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&b\\\ 0&y&z\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{3})=\left\\{\left(\begin{pmatrix}0&0&y\\\ 0&x&z\\\
0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&x&z\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,a\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{4})=\left\\{\left(\begin{pmatrix}2x&y&z\\\ 0&x&0\\\
0&0&x\\\ \end{pmatrix},\begin{pmatrix}0&a&b\\\ 0&x&0\\\ 0&0&x\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{5})=\left\\{\left(\begin{pmatrix}2x&y&t\\\ 0&x&-z\\\
0&z&x\\\ \end{pmatrix},\begin{pmatrix}0&a&b\\\ 0&x&-z\\\ 0&z&x\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,t,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{7}(\alpha))=\left\\{\left(\begin{pmatrix}\gamma
x&y&z\\\ 0&x&\frac{x}{2}\\\ 0&-\frac{x}{2\alpha}&(\gamma-1)x\\\
\end{pmatrix},\begin{pmatrix}0&a&b\\\ 0&x&\frac{x}{2}\\\
0&-\frac{x}{2\alpha}&(\gamma-1)x\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,a,b\in\mathbb{F}\right\\}$, where
$\gamma=\dfrac{4\alpha-1}{2\alpha}$;

  * •

$\operatorname{Bider}(L_{8})=\left\\{\left(\begin{pmatrix}x+y&z&t\\\ 0&x&0\\\
0&0&y\\\ \end{pmatrix},\begin{pmatrix}0&z&t\\\ 0&0&a\\\ 0&0&y\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,t,a\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{9})=\left\\{\left(\begin{pmatrix}x&y&0\\\ y&x&0\\\
0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&0&b\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{10})=\left\\{\left(\begin{pmatrix}x&-y&0\\\ y&x&0\\\
0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&0&b\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{12}(\alpha))=\left\\{\left(\begin{pmatrix}x&\alpha
y&0\\\ y&x+y&0\\\ 0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&0&b\\\
0&0&0\\\ \end{pmatrix}\right)\Bigg{|}x,y,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{13})=\left\\{\left(\begin{pmatrix}x&0&0\\\ 0&y&0\\\
0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&0&b\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{14})=\left\\{\left(\begin{pmatrix}2x&0&y\\\ y&3x&z\\\
0&0&x\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&0&b\\\ 0&0&x\\\
\end{pmatrix}\right)\Bigg{|}x,y,z,a,b\in\mathbb{F}\right\\}$;

  * •

$\operatorname{Bider}(L_{15})=\left\\{\left(\begin{pmatrix}x&0&x\\\ x&0&y\\\
0&0&0\\\ \end{pmatrix},\begin{pmatrix}0&0&a\\\ 0&0&b\\\ 0&0&0\\\
\end{pmatrix}\right)\Bigg{|}x,y,a,b\in\mathbb{F}\right\\}$.

## References

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  * [2] Blokh A., "A generalization of the concept of a Lie algebra", _Doklady Akademii Nauk SSSR_ 165 (1965), no. 3, pp. 471–473.
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colorcitaRGB21,86,130 colorrefRGB5,10,177 colorwebRGB177,6,38

# Frequently recurrence properties and block families

Rodrigo Cardeccia, Santiago Muro  Instituto Balseiro, Universidad Nacional de
Cuyo – C.N.E.A. and CONICET, San Carlos de Bariloche, República Argentina
<EMAIL_ADDRESS>Facultad de Ciencias Exactas, Ingeniería y
Agrimensura, Universidad Nacional de Rosario and CIFASIS, CONICET
<EMAIL_ADDRESS>
###### Abstract.

We study recurrence and the related (weaker form of recurrence)
$\mathscr{P}_{\mathcal{F}}$ property for linear operators on Banach spaces and
for general hereditarily upward families. We show that for special families,
called _block_ families, hypercyclic operators with the
$\mathscr{P}_{\mathcal{F}}$ property are $\mathcal{F}$-hypercyclic.
Reciprocally, if $b\mathcal{F}$ is the block family of a given family
$\mathcal{F}$, adjoint operators that are $b\mathcal{F}$-hypercyclic have the
$\mathscr{P}_{\mathcal{F}}$ property. As a consequence, we are able to answer
an open question about the existence of reiteratively hypercyclic operators.
We also characterize backward shift operators with the
$\mathscr{P}_{\mathcal{F}}$ property.

###### Key words and phrases:

Frequently recurrent operators, Reiteratively hypercylic operators,
Furstenberg families

###### 2010 Mathematics Subject Classification:

47A16, 37B20 37A45 47B37

Partially supported by ANPCyT PICT 2015-2224, UBACyT 20020130300052BA, PIP
11220130100329CO and CONICET

## Introduction

Recurrence is one of the oldest and most studied concepts in the theory of
dynamical systems. In contrast, in the dynamics of linear operators the main
concept is that of hypercyclicity and a systematic study of recurrence began
only around 10 years ago with the work of Costakis, Manoussos and Parissis
[10, 11].

On the other hand, based on the seminal articles [4, 5] much of the interest
in linear dynamics shifted towards the study of $\mathcal{F}$-hypercyclicity
for different hereditarily upward (or Furstenberg) families of subsets of the
natural numbers, hence including in a single theory concepts as hypercyclity,
frequent hypercyclicity, reiterative hypercyclicity or chaos, among others.

Thus at this point, it turns out to be natural to investigate
$\mathcal{F}$-recurrence of linear operators for general Furstenberg families
$\mathcal{F}$. This was recently studied by Bonilla, Grosse-Erdmann, López-
Martínez and Peris [6]. One of their main interests there was to determine
conditions under which an operator which is hypercyclic and
$\mathcal{F}$-recurrent is automatically $\mathcal{F}$-hypercyclic. Our point
of view is slightly different, and a question that we ask is whether
hypercyclicity plus a strong form of recurrence ($\mathcal{F}$-recurrence)
implies a weak form of frequent hypercyclicity ($\mathcal{G}$-hypercyclicity,
for a Furstenberg family $\mathcal{G}$ that is bigger than $\mathcal{F}$).
With this aim in mind we study the block family $b\mathcal{F}$ associated to a
given family $\mathcal{F}$, which is essentially formed by sets that are union
of finite blocks of a given set $A\in\mathcal{F}$, and we show that an
operator that is $\mathcal{F}$-recurrent and hypercyclic is automatically
$b\mathcal{F}$-hypercyclic (see Corollary 2.6). Moreover, this result holds if
in instead of being $\mathcal{F}$-recurrent the operator satisfies a weaker
form of recurrence that is called the $\mathscr{P}_{\mathcal{F}}$ property.
This property turns out to be of vital importance to our investigations.
Indeed, we are able to prove that, for adjoint operators, the converse of the
mentioned result holds: any $b\mathcal{F}$-hypercyclic operator has the
$\mathscr{P}_{\mathcal{F}}$ property (and it is of course hypercyclic), see
Corollary 2.11.

The $\mathscr{P}_{\mathcal{F}}$ property is also important for the
understanding of the dynamics of backward shift operators: in [5], it was
proved that $\mathcal{F}$-hypercyclic unilateral backward shifts are
characterized through two conditions when the family $\mathcal{F}$ is upper.
We show in Section 4 that exactly these two conditions characterize unilateral
backward shifts with the $\mathscr{P}_{\mathcal{F}}$ property for general, non
necessarily upper, families. And we prove that under the assumption of
$\mathcal{F}$ being an upper family, every unilateral backward shift with the
$\mathscr{P}_{\mathcal{F}}$ property is automatically
$\mathcal{F}$-hypercyclic. Moreover, we also show that two very similar
conditions characterize bilateral backward shifts with the
$\mathscr{P}_{\mathcal{F}}$ property, but in this case it is not clear if they
are also $\mathcal{F}$-hypercyclic.

Our investigations on the relationship between the $\mathscr{P}_{\mathcal{F}}$
property and $\mathcal{F}$-hypercyclicity allow us to deduce some properties
of the spectrum of reiteratively hypercyclic operators and to answer (in the
negative, see Corollary 3.7) to the following question, asked in [6, Question
4.10], or in [20, Question 2.6]).

######  Question 0.1.

Does every infinite dimensional separable Banach space support reiteratively
hypercyclic operators?

##  1\. Preliminaries

Given a hereditary upward family $\mathcal{F}\subseteq\mathcal{P}(\mathbb{N})$
(also called Furstenberg family) we say that an operator $T$ on a space $X$ is
$\mathcal{F}$-hypercyclic if there is $x\in X$ for which the sets
$N_{T}(x,U):=\\{n\in\mathbb{N}:T^{n}(x)\in U\\}$ of return times belong to
$\mathcal{F}$. A vector $x$ is $\mathcal{F}$-recurrent for $T$ if
$N_{T}(x,V)\in\mathcal{F}$ for every neighborhood $V$ of $x$. An operator is
called $\mathcal{F}$-recurrent if the set of $\mathcal{F}$-recurrent vectors
is dense. The set of return times $N_{T}(x,U)$ will be denoted by $N(x,U)$
when the operator is from the context.

Thus, for example, if we take $\mathcal{F}_{\infty}$ to be the family of
infinite sets, $\mathcal{F}_{\infty}$-hypercyclicity and
$\mathcal{F}_{\infty}$-recurrence are simply hypercyclity and recurrence
respectively.

Let us recall some examples of families, which are the most widely studied in
the literature:

  * •

$\underline{\mathcal{D}}=\\{$ sets with positive lower density$\\}$ (i.e.
$A\in\underline{\mathcal{D}}$ if $\underline{d}(A):=$
$\liminf_{n}\\#\frac{\\{k\leq n:k\in A\\}}{n}>0$). An operator is frequently
hypercyclic if and only if it is $\underline{\mathcal{D}}$\- hypercyclic.

  * •

$\overline{\mathcal{D}}=\\{$ sets with positive upper density$\\}$ (i.e.
$A\in\overline{\mathcal{D}}$ if $\overline{d}(A):=$
$\limsup_{n}\\#\frac{\\{k\leq n:k\in A\\}}{n}>0$). An operator is upper
frequently hypercyclic if and only if it is $\overline{\mathcal{D}}$\-
hypercyclic.

  * •

$\overline{\mathcal{BD}}=\\{$ sets with positive Banach upper density$\\}$
(i.e. $A\in\overline{\mathcal{BD}}$ if
$\overline{Bd}(A):=\lim_{n}\sup_{k}\\#\frac{\\#A\cap[k,k+n]}{n}>0$). An
operator is reiterative hypercyclic if and only if it is
$\overline{\mathcal{BD}}$\- hypercyclic. The family $\underline{\mathcal{BD}}$
is defined analogously.

  * •

$\overline{\mathcal{LD}}=\\{\text{sets with upper logarithmic density}\\}$,
i.e sets such that $\overline{\delta}(A):=\limsup_{N}\frac{1}{\log
N}\sum_{j\leq N,j\in A}\frac{1}{j}>0$. The family $\underline{\mathcal{LD}}$
is defined analogously.

Given a set of natural numbers $A$, the following relation holds (see e.g.
[25, Part III, Theorem 1.2]):

(1)
$\underline{Bd}(A)\leq\underline{d}(A)\leq\underline{\delta}(A)\leq\overline{\delta}(A)\leq\overline{d}(A)\leq\overline{Bd}(A).$

Thus, if a set has natural density, then it has logarithmic density.

  * •

$Thick=\\{\text{ sets with arbitrarily long intervals }\\}$.

  * •

$\mathcal{S}yn=\\{\text{ Syndetic sets, i.e. they have bounded gaps }\\}$.
Equivalently, $A\in\mathcal{S}yn$ if there is a finite subset $F$ of $S$ such
that $\bigcup_{n\in F}S-n=\mathbb{N}$.

  * •

$\mathcal{PS}yn=\\{$ Piecewise syndetic sets, i.e. there is $b$ such that $A$
supports arbitrarily long subsets with gaps bounded by $b$ $\\}$.
Equivalently, $A\in\mathcal{PS}yn$ if there is a finite set $F$ such that

$\bigcup_{n\in F}A+n\text{ is thick.}$

  * •

$\mathcal{AP}\\!_{b}$=$\\{$ sets with arbitrarily long arithmetic progressions
with bounded step$\\}$.

A hereditary upward family $\mathcal{F}$ is said to be shift invariant if it
satisfies that $A\in\mathcal{F}$ if and only if $A+k\in\mathcal{F}$ for any
$k\in\mathbb{Z}$ and it is said to be finitely invariant provided that for
every $A\in\mathcal{F}$ and every $k\in\mathbb{N}$, we have that
$[k,\infty)\cap A\in\mathcal{F}$. All families mentioned above are shift
invariant and finitely invariant.

A hereditary upward family is said to be upper provided that
$\emptyset\notin\mathcal{F}$ and $\mathcal{F}$ can be written as

$\bigcup_{\delta\in D}\mathcal{F}_{\delta},\quad\textrm{with
}\;\mathcal{F}_{\delta}=\bigcap_{m\in M}\mathcal{F}_{\delta,m},$

where $D$ is arbitrary but $M$ is countable and such that the families
$\mathcal{F}_{\delta,m}$ and $\mathcal{F}$ satisfy

  * •

each $\mathcal{F}_{\delta,m}$ is finitely hereditary upward, that means that
for each $A\in\mathcal{F}_{\delta,m}$, there is a finite set $F$ such that
$F\cap A\subseteq B,$ then $B\in\mathcal{F}_{\delta,m}$;

  * •

$\mathcal{F}$ is uniformly left invariant, that is, if $A\in\mathcal{F}$ then
there is $\delta\in D$ such that for every $n$, $A-n\in\mathcal{F}_{\delta}$.

The families $\mathcal{F}_{\infty}$,
$\overline{\mathcal{D}},\overline{\mathcal{B}D}$ are upper while
$\underline{\mathcal{D}}$ is not upper (see [5]).

We finish this preliminaries with some equivalences for
$\mathcal{F}$-hypercyclicity for upper families (for more
$\mathcal{F}$-hypercyclicity see [4, 5, 6, 8, 12, 22, 23, 24]).

######  Theorem 1.1 (Bonilla-Grosse Erdmann [5]).

Let $\mathcal{F}$ be an upper hereditary upward family and let $T$ be a linear
operator on a separable Fréchet space. Then the following are equivalent:

  1. (1)

$T$ is $\mathcal{F}$-hypercyclic.

  2. (2)

For any nonempty open set $V$ there is $\delta\in D$ such that for any
nonempty open set $U$ there is $x\in U$ with
$N_{T}(x,V)\in\mathcal{F}_{\delta}.$

  3. (3)

For any nonempty open set $V$ there is $\delta\in D$ such that for every $U$
and $m\in M$ there is $x\in U$ with $N_{T}(x,V)\in\mathcal{F}_{\delta,m}$.

  4. (4)

The set of $\mathcal{F}$-hypercyclic points is residual.

##  2\. Block families, $\mathcal{F}$-hypercyclicity and the
$\mathscr{P}_{\mathcal{F}}$ property

The following notion of recurrence was formally defined in [23]. For
particular families, it has been considered by several authors (see for
instance [11], [2, Proposition 4.6] and [16, Section 2.5]).

######  Definition 2.1.

Let $T$ be a linear operator and let $\mathcal{F}$ be a Furstenberg family. We
will say that $T$ has the $\mathscr{P}_{\mathcal{F}}$ property provided that
for every nonempty open set $U$, there is $x\in X$ such that
$N(x,U)\in\mathcal{F}$.

Note that if $\mathcal{F}$ is a shift invariant family then $T$ has the
$\mathscr{P}_{\mathcal{F}}$ property if and only if for every nonempty open
set $U$, there is $x\in X$ such that $N(x,U)\in\mathcal{F}$. The
$\mathscr{P}_{\mathcal{F}}$ property is weaker than $\mathcal{F}$-recurrence
(at least formally, see Section 5)

We will be specially interested in the $\mathscr{P}_{\mathcal{F}}$ property
for a special class of families called block families (see [14], see also [18,
19]).

######  Definition 2.2.

Given a Furstenberg family $\mathcal{F}$ we define the block family
$b\mathcal{F}$ in the following way: $S\in b\mathcal{F}$ if and only if there
is $F\in\mathcal{F}$ such that for every finite subset $R$ of $F$ there is
$n\in\mathbb{N}$ such that $R+n\subseteq S$.

######  Proposition 2.3 (Basic properties).

Let $\mathcal{F},\tilde{\mathcal{F}}$ be Furstenberg families, then

  1. (1)

$\mathcal{F}\subseteq b\mathcal{F}$;

  2. (2)

$bb\mathcal{F}=b\mathcal{F}$;

  3. (3)

If $\mathcal{F}\subseteq\tilde{\mathcal{F}}$ then $b\mathcal{F}\subseteq
b\tilde{\mathcal{F}}$;

  4. (4)

$b\mathcal{F}\subseteq b\mathcal{F}+n$ for every $n$

###### Proof.

(1) and (3) are immediate. Proof of (2). By (1) we have that
$b\mathcal{F}\subseteq bb\mathcal{F}$ so that one inclusion holds.

Let $S\in bbF$, hence there is $F_{b}\in b\mathcal{F}$ such that for every
finite subset $R_{b}$ of $b\mathcal{F}$ there is $n$ such that
$R_{b}+n\subseteq S$. Also, since $F_{b}\in b\mathcal{F}$, there is
$F\in\mathcal{F}$ such that for every finite subset $R$ of $F$ there is $n$
such that $R+n\subseteq F_{b}$. Therefore, if $R$ is a finite subset of $F$
and $R+n\subseteq F_{b}$, there is $n^{\prime}$ such that
$R+n+n^{\prime}\subseteq S$, which proves that $S\in b\mathcal{F}$.

Proof of $(4)$. Let $S\in b\mathcal{F}$ and $R\in\mathcal{F}$ such that for
every finite set $F$ of $R$ there is $k$ such that $F+k\subseteq S$. Thus,
$F+k+n\subseteq S$.

∎

We provide next some examples of block families.

######  Example 2.4.

  1. a)

Let $\mathcal{F}_{\infty}$ and $\mathcal{C}ofin$, where $\mathcal{C}ofin$
denotes the family of cofinite subsets. Then,
$b\mathcal{F}_{\infty}=\mathcal{F}_{\infty}$ and $b\mathcal{C}ofin=Thick$. In
particular $bThick=Thick$.

  2. b)

$b\mathcal{D}$=$b\underline{\mathcal{D}}=b\overline{\mathcal{D}}=\overline{\mathcal{BD}}$.

###### Proof.

One inclusion is Ellis’ Theorem [13, Theorem 3.20], which says precisely that
if $A\in\mathcal{BD}$, then there is $F$ of positive density such that for
every finite subset $R$ of $F$ there is $n$ with $R+n\in A$. Thus,
$\overline{\mathcal{BD}}\subseteq b\mathcal{D}$.

For the converse, let $A\in b\overline{\mathcal{D}}$. Hence there is $F\in
b\overline{\mathcal{D}}$ such that for every finite subset $R$ of $F$ there is
$n\in\mathbb{N}$ such that $F+n\subseteq A$. As $F$ has upper density, there
are $\delta>0$ and $n_{k}$ such that $\frac{\\#\\{k\leq n_{k}:k\in
F\\}}{n_{k}}>\delta$ for every $k$.

Let $R_{k}=\\{k\leq n_{k}:k\in F\\}$, hence
$\frac{\\#R_{k}\cap[0,n_{k}]}{n_{k}}>\delta.$ If $R+n\subseteq A$ it follows
that $\frac{\\#A\cap[n,n+n_{k}]}{n_{k}}>\delta$. We conclude that $A$ has
upper Banach density. Thus, $b\mathcal{D}\subseteq
b\underline{\mathcal{D}}\subseteq
b\overline{\mathcal{D}}\subseteq\overline{\mathcal{BD}}$. ∎

  3. c)

$b\mathcal{LD}=b\underline{\mathcal{LD}}=b\overline{\mathcal{LD}}=\overline{\mathcal{BD}}.$

###### Proof.

Let $A\in\overline{\mathcal{BD}}$. By $b)$, there is a set $R$ with positive
density such that for every finite subset $F$ of $R$ there is $n$ such that
$F+n\subseteq A$. By (1), we have that $R$ has positive logarithmic density.
This implies that $A\in{b\mathcal{LD}}.$ Conversely, if $A\in
b\overline{\mathcal{LD}}$ then, since
$\overline{\mathcal{LD}}\subseteq\overline{\mathcal{D}}$, we have that $A\in
b\overline{\mathcal{D}}=\overline{\mathcal{BD}}.$ This proves our claim since
$\mathcal{LD}\subseteq\underline{\mathcal{LD}}\subseteq\overline{\mathcal{LD}}$.
∎

  4. d)

$b\mathcal{S}yn=\mathcal{PS}yn$.

###### Proof.

Let $A\in b\mathcal{S}yn$. Then there is $F\in\mathcal{S}yn$ such that for
every finite set $R\subseteq F$ there is $n\in\mathbb{N}$ such that
$R+n\subseteq A$. Let $b$ a bound for the gaps of $F$ and consider for each
$k$, a finite set $R_{k}\subseteq F$ such that $|R_{k}|=k$ and $n_{k}$ such
that $R_{k}+n_{k}\subseteq A$. Hence,
$\bigcup_{i\in[0,b]}R_{k}+n_{k}+i\subseteq\bigcup_{i\in[0,b]}A+i$ contains an
interval of length at least $k$. This implies that $\bigcup_{i\in[0,b]}A+i$ is
thick. Equivalently, $A\in\mathcal{PS}$.

Let $A\in\mathcal{PS}yn$. Then there are $b>0,(n_{m})_{m}$ and
$(a_{m,k})_{m,1\leq k\leq m}$ such that

    1. i)

for every $m$ and $1\leq k\leq m$, $n_{m}+a_{m,1}+\ldots+a_{m,k}\in A$ and

    2. ii)

for every $m$ and $k\leq m$, $a_{m,k}\in[1,b]$.

The sequence $(a_{m,1})_{m}$ is infinite and belongs to $[1,b]$. Hence, there
is $f_{1}\in[1,b]$ such that $a_{m,1}=f_{1}$ for infinitely many
$m^{\prime}s$. Consider the subsequences of $(n_{m})_{m}$ and $(a_{m,k})_{m}$,
which we keep notating $(n_{m})_{m}$ and $(a_{m,k})_{m}$ respectively.

Thus,

    1. i)

for every $m$, $n_{m}+f_{1}\in A$;

    2. ii)

for every $m$ and $2\leq k\leq m$, $n_{m}+f_{1}+a_{m,2}+\dots+a_{m,k}\in A$;

    3. iii)

$f_{1}\in[1,b]$ and

    4. iv)

for every $m$ and $2\leq k\leq m$, $a_{m,k}\in[1,b]$.

Proceeding by induction we construct for each $m$, $f_{m}\in[1,b]$ such that
for every $k\leq m$,

$n_{m}+f_{1}+\ldots+f_{k}\in A.$

Let $F=\\{\sum_{i=1}^{m}f_{i}:m\in\mathbb{N}\\}$. Thus $F$ is syndetic and for
every finite set $R\subseteq F$ there is $n$ such that $R+n\subseteq A$. ∎

  5. e)

$b\mathcal{IAP}=\mathcal{AP}\\!_{b}$, where $\mathcal{IAP}$ is the family of
natural numbers supporting an infinite arithmetic progression.

In [6, Theorem 8.4] it was proved that for upper right-invariant families, an
operator is $\mathcal{F}$-hypercyclic if and only if it is hypercyclic and has
a residual set of $\mathcal{F}$-recurrent vectors. The next result shows, in
particular, that for any block family, the residuality of
$\mathcal{F}$-recurrent vectors is not needed. For the family of sets with
positive upper Banach density the following proposition also answers [23,
Question 23].

######  Proposition 2.5.

Let $T$ be an operator and let $\mathcal{F}$ be a block family. Then $T$ is
$\mathcal{F}$-hypercyclic if and only if $T$ is hypercyclic and has the
$\mathscr{P}_{\mathcal{F}}$ property.

###### Proof.

If $T$ is $\mathcal{F}$-hypercyclic then it is clear that $T$ has the
$\mathscr{P}_{\mathcal{F}}$ property.

Suppose now that $T$ is hypercyclic and that it has the
$\mathscr{P}_{\mathcal{F}}$ property. Let $x$ be a hypercyclic vector and $U$
be a nonempty open set. Let $y\in X$ such that $A=N(y,U)\in\mathcal{F}$.

Let $R$ be a finite subset of $A$. Thus, $W=\bigcap_{r\in R}T^{-r}(U)$ is an
open set containing $y$. Since $x$ is a hypercyclic vector there is $n$ that
$T^{n}(x)\in W$ and hence $R+n\subseteq N(x,U)$. Thus, $N(x,U)\in
b\mathcal{F}=\mathcal{F}$ and $x$ is an $\mathcal{F}$-hypercyclic vector. ∎

######  Corollary 2.6.

Let $T$ be an operator and let $\mathcal{F}$ be a Furstenberg family. If $T$
is hypercyclic and has the $\mathscr{P}_{\mathcal{F}}$ property, then $T$ is
$b\mathcal{F}$-hypercyclic.

We will see in Theorem 2.10 that the converse holds for adjoint operators.

It was proved in [12, Theorem 2.5] that, for some upper Furstenberg families
$\mathcal{F}$, if $T$ is $\mathcal{F}$-hypercyclic then $T\oplus T$ is also
$\mathcal{F}$-hypercyclic. We show now as a consequence of the Proposition 2.5
that this is also true for block families.

######  Proposition 2.7.

Let $T$ be an operator such that its commutant contains an universal vector,
i.e. there is some $x\in X$ such that the set $\\{Sx\,:\,ST=TS,\,S:X\to
X\,\textrm{continuous mapping}\\}$ is dense in $X$.

Then we have the following.

  1. a)

If $T$ has the $\mathscr{P}_{\mathcal{F}}$ property then
$\bigoplus_{i=1}^{n}T$ has the $\mathscr{P}_{\mathcal{F}}$ property.

  2. b)

If $T$ is $\mathcal{F}$-recurrent then $\bigoplus_{i=1}^{n}T$ is
$\mathcal{F}$-recurrent.

  3. c)

If $\mathcal{F}$ is a block family and $T$ is weakly mixing then
$\bigoplus_{i=1}^{n}T$ is $\mathcal{F}$-hypercyclic.

In particular, the above statements hold if $T$ is cyclic or if it commutes
with a cyclic operator.

We note that a similar result was proved in [10, Theorem 9.1] for
$\mathcal{F}=\mathcal{F}_{\infty},$ the family of infinite subsets, with the
additional hypothesis that the set of all vectors $x\in X$ such that
$\overline{\\{Sx\,:\,ST=TS,\,S\in\mathcal{L}(X)\\}}=X$ is residual.

###### Proof.

Proof of $a)$. Let $V_{1}\ldots V_{n}$ be nonempty open sets and let $v\in X$
be a universal vector for the commutant of $T$. Hence, there are mappings
$S_{i}$ that commute with $T$ and a nonempty open set $W$ around $v$ such that
$S_{i}(W)\subseteq V_{i}$ for every $1\leq i\leq n$. Let $x\in W$ be such that
$N_{T}(x,W)\in\mathcal{F}$. Then,
$\oplus_{i=1}^{n}S_{i}(x)\in\bigoplus_{i=1}^{n}V_{i}$ and if $k\in N_{T}(x,W)$
then $T^{k}S_{i}(x)=S_{i}T^{k}(x)\in S_{i}(W)\subseteq V_{i}$. Therefore,
$N_{T}(x,W)\subseteq\bigcap_{i=1}^{n}N_{T}(S_{i}(x),V_{i})$, which implies
that
$N_{\bigoplus_{i=1}^{n}T}\left(\oplus_{i=1}^{n}S_{i}(x),\oplus_{i=1}^{n}V_{i}\right)\in\mathcal{F}$.

Proof of $b)$. Let $V_{1}\ldots V_{n}$ be nonempty open sets and let $v\in X$
be a universal vector for the commutant. Hence, there are mappings $S_{i}$
that commute with $T$ and a nonempty open set $W$ around $v$ such that
$S_{i}(W)\subseteq V_{i}$ for every $1\leq i\leq n$. Let $x\in W$ be an
$\mathcal{F}$-recurrent vector, thus $N_{T}(x,W)\in\mathcal{F}$. Then,
$\oplus_{i=1}^{n}S_{i}(x)\in\bigoplus_{i=1}^{n}V_{i}$.

We show that $\oplus_{i=1}^{n}S_{i}(x)$ is an $\mathcal{F}$-recurrent vector
for $\bigoplus_{i=1}^{n}T$. Let $U_{i}$ be open neighbourhoods of $S_{i}(x),$
$i=1,\dots,n$. We must show that
$\bigcap_{i=1}^{n}N(S_{i}(x),U_{i})\in\mathcal{F}$. We may assume that
$U_{i}\subseteq V_{i}$ for every $i.$ Then since $S_{i}(x)\in S_{i}(W)\cap
U_{i},$ there are non-empty open sets $W_{i}$ such that $x\in W_{i}\subseteq
W$ and $S_{i}(W_{i})\subseteq S_{i}(W)\cap U_{i}$ for every $i=1,\dots,n$. Let
$\tilde{W}:=\bigcap_{i=1}^{n}W_{i}$. Then $S_{i}(\tilde{W})\subseteq U_{i}$
for every $i$ and thus $N_{T}(x,\tilde{W})\subseteq N_{T}(S_{i}(x),U_{i})$.
Since $N_{T}(x,\tilde{W})\in\mathcal{F}$ we conclude that
$\bigcap_{i=1}^{n}N_{T}(S_{i}(x),U_{i})\in\mathcal{F}$.

Proof of $c)$. By $a)$ $\bigoplus_{i=1}^{n}T$ has the
$\mathscr{P}_{\mathcal{F}}$ property. Since $\bigoplus_{i=1}^{n}T$ is
hypercyclic, it follows by Proposition 2.5 that $\bigoplus_{i=1}^{n}T$ is
$\mathcal{F}$-hypercyclic. ∎

Most commonly used block upper families, like $\mathcal{F}=\mathcal{PS}yn$,
$\mathcal{AP}\\!_{b}$ or $\overline{\mathcal{BD}}$, satisfy the following
invariance property:

(2) $\displaystyle\textrm{ if }F\in\mathcal{F}_{\delta,m}\textrm{ and
}k\in\mathbb{N},\textrm{ then }F+k\in\mathcal{F}_{\delta,m}.$

Note that this is not true for $\underline{\mathcal{D}}$ and
$\overline{\mathcal{D}}$.

The following proposition is similar to Theorem 1.1 but simpler as the proof
does not rely on Baire’s Category Theorem.

######  Proposition 2.8.

Let $\mathcal{F}$ be an upper family with the invariance property (2). Then
the following are equivalent.

  1. (1)

Every hypercyclic vector is an $\mathcal{F}$-hypercyclic vector;

  2. (2)

there is an $\mathcal{F}$-hypercyclic vector;

  3. (3)

$T$ is hypercyclic and for every nonempty open set $U$ there is $\delta>0$
such that for every $m$ there is $x\in U$ such that
$N(x,U)\in\mathcal{F}_{\delta,m}$.

We will need the following simple lemma for its proof.

######  Lemma 2.9.

Let $\mathcal{F}$ be an upper family satisfying property (2). Then
$A\in\mathcal{F}$ if and only if there are $\delta$, $k_{m}$ and finite sets
$R_{m}\in\mathcal{F}_{\delta,m}$ such that for every $m$

$k_{m}+R_{m}\subseteq A.$

###### Proof.

Let $A\in\mathcal{F}$. Then there is $\delta>0$ such that
$A\in\mathcal{F}_{\delta}=\bigcap_{m}F_{\delta,m}$. Since each
$\mathcal{F}_{\delta,m}$ is finitely hereditary upward, there are finite sets
$R_{m}\in\mathcal{F}_{\delta,m}$ such that $B\in\mathcal{F}_{\delta,m}$
whenever $A\cap R_{m}\subseteq B$. Thus, $A\cap
R_{m}\in\mathcal{F}_{\delta,m}$ and $A\cap R_{m}\subseteq A.$

Reciprocally let $A\subseteq\mathbb{N}$ such that there are $\delta>0$,
$k_{m}$ and finite sets $R_{m}\in\mathcal{F}_{\delta,m}$ with
$k_{m}+R_{m}\subseteq A$ for every $m$. Note that
$k_{m}+R_{m}\in\mathcal{F}_{\delta,m}$ for every $m$.

Let $R=\bigcup_{m\in\mathbb{N}}k_{m}+R_{m}$. Then,
$R\in\bigcap_{m}\mathcal{F}_{\delta,m}=\mathcal{F}_{\delta}\subseteq\mathcal{F}$.
Since $R\subseteq A,$ we conclude that $A\in\mathcal{F}$. ∎

###### Proof of Proposition 2.8..

We only prove (3) $\Rightarrow(1)$, being the other implications immediate.

Let $x\in X$ be a hypercyclic vector and $U$ be a nonempty open set. There
exists $\delta$ such that for every $m$, there is $y_{m}\in U$ such that
$N(y_{m},U)\in\mathcal{F}_{\delta,m}$. Since $\mathcal{F}_{\delta,m}$ is
finitely hereditary upward, there exist for each $m$ a finite set
$R_{m}\subseteq N(y_{m},U)$ such that if $R_{m}\subseteq A$ then
$A\in\mathcal{F}_{\delta,m}$. Thus, for every $m$, $\bigcap_{r\in
R_{m}}T^{-r}(U)\neq\emptyset$.

Let $k_{m}$ such that $T^{k_{m}}(x)\in\bigcap_{r\in R_{m}}T^{-r}(U).$ Thus,
$k_{m}+R_{m}\subseteq N(x,U)$. By the above lemma this implies that
$N(x,U)\in\mathcal{F}.$ ∎

The following is the main result of this section. It establishes the converse
of Corollary 2.6 for adjoint operators.

######  Theorem 2.10.

Let $T$ be a $w^{*}$-$w^{*}$ continuous operator defined on a dual Banach
space and $\mathcal{F}$ be a Furstenberg family. If $T$ has the
$\mathscr{P}_{b\mathcal{F}}$ property then $T$ has the
$\mathscr{P}_{\mathcal{F}}$ property.

###### Proof.

Let $U$ be a nonempty open set. Let $V\subseteq\overline{V}\subseteq U$ be an
auxiliary ball, so that $\overline{V}$ is $w^{*}$-compact and convex. Let
$x\in V$ such that $N(x,V)\in b\mathcal{F}$. Thus, there is $R\in\mathcal{F}$
such that for every finite subset $F$ of $R$ there is $n$ such that
$F+n\subseteq N(x,V)$.

Let $R_{n}=R\cap[0,n]$ and $a_{n}$ such that $T^{a_{n}+r}(x)\in U$ for every
$r\in R_{n}$. Since $\overline{V}$ is weakly∗\- compact and
$(\overline{V},w^{*})$ is metrizable, there is a subsequence $a_{n_{k}}$ and
$y\in\overline{V}$ such that $T^{a_{n_{k}}}(x)\to y$. We claim that
$R\subseteq N(y,U)$. Indeed, given $r\in R$,
$T^{r}(y)=T^{r}(w^{*}-\lim_{k}T^{a_{n_{k}}}(x))=w^{*}-\lim_{k}T^{a_{n_{k}}+r}(x)$
and $T^{a_{n_{k}}+r}(x)\in\overline{V}\subseteq U$ provided that $r\in R_{k}$.
∎

The above result is no longer true for non $w^{*}$-$w^{*}$ continuous
operators (see Example 4.7).

Theorem 2.10, together with Corollary 2.6, gives us the following.

######  Corollary 2.11.

Let $T$ be a $w^{*}$-$w^{*}$ continuous operator defined in a dual Banach
space and let $\mathcal{F}$ be a Furstenberg family. Then $T$ is hypercyclic
and has the $\mathscr{P}_{\mathcal{F}}$ property if and only if $T$ is
$b\mathcal{F}$-hypercyclic.

######  Corollary 2.12.

Let $T$ be a $w^{*}$-$w^{*}$ continuous operator on a dual Banach space $X$.
Then,

  1. (1)

$T$ has the $\mathscr{P}_{\mathcal{D}}$ property if and only if it has the
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property;

  2. (2)

$T$ has the $\mathscr{P}_{\mathcal{S}yn}$ property if and only if it has the
$\mathscr{P}_{\mathcal{PS}yn}$ property;

  3. (3)

$T$ has a dense set of periodic vectors if and only if it has the
$\mathscr{P}_{\mathcal{AP}\\!_{b}}$ property.

###### Proof.

We only have to prove (3), since (1) and (2) are partcular cases of Theorem
2.10. Since by the above result $T$ has the
$\mathscr{P}_{\mathcal{I}\mathcal{AP}}$ property if and only if it has the
$\mathscr{P}_{\mathcal{AP}\\!_{b}}$ property, it suffices to see that if $T$
has the $\mathscr{P}_{\mathcal{I}\mathcal{AP}}$ property then $T$ has a dense
set of periodic vectors. Given a nonempty open set $U$ take another nonempty
open, bounded and convex set $V\subseteq\overline{V}\subseteq U$. Then there
are $x_{V}\in V$ and $k\in\mathbb{N}$ such $Orb_{T^{k}}(x_{V})\subseteq V$.
Let $Y:=\overline{co}(Orb_{T^{k}}(x_{V}))\subseteq\overline{V}\subseteq U$.
Then $Y$ is a $T^{k}$-invariant $w^{*}$-compact set. Thus by the Schauder-
Tychonoff Theorem, there is a fixed point of $T^{k}$ in $Y$. This fixed point
is a $k$-periodic vector for $T$ which is contained in $U$. ∎

The following result is similar to Theorem 2.10 and will be needed in the next
section. We prove it for the sake of the completeness.

######  Theorem 2.13.

Let $X$ be a Banach space, $T$ be a linear operator and $\mathcal{F}$ be a
Furstenberg family. If $T$ has the $\mathscr{P}_{b\mathcal{F}}$ property then
for each nonempty open set $V\subseteq X^{**}$ such that $V\cap
X\neq\emptyset$, there is $z\in V$ such that $N_{T^{**}}(z,V)\in\mathcal{F}$.

###### Proof.

Let $V\subseteq X^{**}$ be a nonempty set such that $V\cap X\neq\emptyset.$
Consider $U$ an open ball of $X$ such that $U\subseteq V\cap X$ and
$\overline{U}\subseteq V$, where we see $\overline{U}$ as a closed ball of
$X^{**}$. Therefore, $\overline{U}$ is a $w^{*}$-compact and metrizable subset
of $X^{**}$.

Let $x\in U$ such that $N(x,U)\in b\mathcal{F}$ and $R\in\mathcal{F}$ such
that for every finite subset $F$ of $R$ there is $n$ such that $F+n\in R$. Let
$R_{n}=R\cap[0,n]$ and $a_{n}$ such that $R_{n}+a_{n}\subseteq N(x,U)$. Since
$\overline{U}\subseteq X^{**}$ is $w^{*}$-compact and metrizable there is
$z\in\overline{U}\subseteq V$ and a subsequence $a_{n_{k}}$ such that
$(T^{**})^{a_{n_{k}}}(x)\to z$.

We will prove that $R\subseteq N(y,V)$. Let $r\in R$. Then,
$(T^{**})^{r}(z)=(T^{**})^{r}(w^{*}-limT^{a_{n_{k}}}(x))=w^{*}-\lim(T^{**})^{a_{n_{k}}+r}(x)$.
On the other hand, $(T^{**})^{a_{n_{k}}+r}(x)\in U$ provided that $k>r$.
Therefore $(T^{**})^{r}(y)\in\overline{U}\subseteq V$. ∎

We finish this section with some comments about the existence of
$\mathcal{F}$-recurrent operators for two particular families.

In [4, Section 3] it is proved that if $\mathcal{F}$ is contained in the
family of sets of positive lower Banach density, i.e. in the family of
syndetic sets then there are no $\mathcal{F}$-hypercyclic operators. A similar
result for recurrence does not hold because there are non-trivial examples of
$\mathcal{F}$-recurrent operators. For example, every chaotic operator is
$\mathcal{S}yn$-recurrent and in [6, Corollary 4.7], it is also shown that
there are hypercyclic $\mathcal{S}yn$-recurrent operators without periodic
points.

On the other hand it is also proved in [4, Section 3] that if $\mathcal{F}$ is
contained in the family of sets of upper Banach density $\geq\delta$ then
there are no $\mathcal{F}$-hypercyclic operators. We will see next that there
is a generalization of this result, in the sense that the only
$\mathcal{F}$-recurrent operators are the trivial ones (multiples of the
identity).

######  Lemma 2.14.

Let $T$ be a continuous mapping, $V$ a nonempty open set and $\delta>0$ such
that for every nonempty open set $U\subseteq V$ there is $x_{U}\in U$ such
that $\overline{Bd}(N(x_{U},U))>\delta$. Then, there is $j$ such that
$T^{j}|_{V}=Id.$

###### Proof.

Let us proceed by contradiction. We may assume that $V$ is connected. Let
$k<\delta^{-1}$. We notice that for every open set $U\subseteq V$ there is
$j\leq k$ such that $T^{j}(U)\cap U\neq\emptyset$. Indeed, there are $x_{U}\in
U$, $n$ and $n+j$ with $j\leq k$ such that $\\{n,n+j\\}\subseteq N(x_{U},U)$.
Therefore $T^{n}(x_{U})\in U$ and $T^{j}(T^{n}(x_{U}))\in U$.

Suppose that there exists $x\in V$ such that $T^{j}(x)\neq x$ for every $j\leq
k$. Thus there are $V_{j}$ open neighbourhoods of $T^{j}x$ such that
$x\notin\overline{V_{j}}$, for $j=1,\dots,k.$ By continuity, there is some
open neighbourhood $V_{0}\subseteq V$ of $x$ such that $T^{j}(V_{0})\subseteq
V_{j}$ and $V_{0}\cap V_{j}=\emptyset$ for every $1\leq j\leq k$. This is a
contradiction, because it implies that $T^{j}(V_{0})\cap V_{0}=\emptyset$ for
$j=1,\dots,k.$

This implies that every vector in $V$ is periodic of period $\leq k$. Since
the set of periodic vectors of a given period is closed, the connectivity of
$V$ implies that there is at least one $j\leq k$ such that the set of periodic
vectors of period $j$ is $V$, i.e. $T^{j}|_{V}=id$. ∎

######  Corollary 2.15.

If $\mathcal{F}$ is contained in the family of sets of upper Banach density
$\geq\delta$, then the only operators with the $\mathscr{P}_{\mathcal{F}}$
property are of the form $\lambda Id$, with $\lambda^{k}=1$ and
$k\leq\delta^{-1}$.

##  3\. Spectrum of reiteratively hypercyclic operators

In this section we prove that operators having the
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property that are not unimodular
multiples of the identity have perfect spectrum. Thus, reiteratively and
piecewise syndetic hypercyclic operators have perfect spectrum. This will
allow us to provide an answer to Question 0.1.

The following Lemma was proved in [8]. Its proof is a slight modification of
an analogous result of Shkarin for frequently hypercyclic operators (see [24]
or [17, Lemma 9.38]).

######  Lemma 3.1.

Let $S$ be an operator, $x^{*}\in X^{*}\setminus\\{0\\}$ and
$U=\\{y:Re(\left\langle y,x^{*}\right\rangle)>0,Re(\left\langle
S(y),x^{*}\right\rangle)<0\\}$. Suppose that for some $x\in U\setminus
ker(x^{*})$,

$\underline{d}\big{(}N_{S}(x,U)\big{)}>0.$

Then $S-I$ is not quasinilpotent.

Recall that a Furstenberg family $\mathcal{F}$ is said to have the CuSP
property provided that for any covering $I_{1}\ldots I_{q}$ of $\mathbb{N}$
and any numbers $n_{1},\ldots n_{q}$ then

$A\in\mathcal{F}\Rightarrow\bigcup_{j=1}^{q}n_{j}+A\cap I_{j}\in\mathcal{F}.$

######  Proposition 3.2.

Let $\mathcal{F}$ be a Furstenberg family with the CuSP property. If $T$ has
the $\mathscr{P}_{\mathcal{F}}$ property then for any $|\lambda|=1$, $\lambda
T$ has the $\mathscr{P}_{\mathcal{F}}$ property.

###### Proof.

We follow the ideas from [6].

Since $T$ has property $\mathscr{P}_{\mathcal{F}}$ it has
$\mathscr{P}_{\mathcal{F}_{\infty}}$ property and by [10, Proposition 2.1] it
is also recurrent. Let $U$ be a nonempty open set and $y\in U$ be a recurrent
vector for $T$. Let $U_{k}\cap U$ be a decreasing chain of open sets contained
in $U$ and converging to $y$. We define

$\Lambda_{k}=\\{\mu\in\mathbb{T}:\exists\ n\text{ and }U_{l}\subseteq
U_{k}\text{ such that }\lambda^{n}=\mu\text{ and }T^{n}(U_{l})\subseteq
U_{k}\\}.$

Since the $\Lambda_{k}$ are decreasing,
$\Lambda=\bigcap_{k\in\mathbb{N}}\overline{\Lambda_{k}}$ is a nonempty and
closed subset of the unit circle.

We see now that $\Lambda$ is a multiplicative semigroup of the unit circle.

Let $\mu_{1},\mu_{2}\in\Lambda$. Let $k\in\mathbb{N}$ and $\varepsilon>0$.
There are numbers $l_{1}$ and $n_{l_{1}}$ such that
$|\lambda^{n_{l_{1}}}-\mu_{1}|<\varepsilon$ and such that
$T^{n_{l_{1}}}(U_{l_{1}})\subseteq U_{k}$. Also, there are $l_{2}>l_{1}$ and
$n_{l_{2}}$ such that $T^{n_{l_{2}}}(U_{l_{2}})\subseteq U_{l_{1}}$, and
$|\lambda^{n_{l_{2}}}-\mu_{2}|<\varepsilon$. Altogether we find that
$T^{n_{l_{1}}+n_{l_{2}}}(U_{l_{2}})\subseteq U_{k}$. Hence,
$\lambda^{n_{l_{1}}}\lambda^{n_{l_{2}}}\in\Lambda_{k}$. Finally,
$|\mu_{1}\mu_{2}-\lambda^{n_{l_{1}}}\lambda^{n_{l_{2}}}|<2\varepsilon$. Hence
$\mu_{1}\mu_{2}\in\overline{\Lambda_{k}}$ for every $k$.

Since $\Lambda$ is a closed multiplicative subgroup of $\mathbb{T}$, then:
either $\Lambda=\mathbb{T}$ or $\Lambda=G_{n}$ for some $n$ (the group of
$n$-th roots of the unity).

We start by analysing the case $\Lambda=\mathbb{T}$. Fix $k\geq 1$ and
$\varepsilon>0$ such that $B(1,\varepsilon)U_{k}\subseteq U$. Since
$\Lambda=\mathbb{T}$, we have that $\Lambda_{k}$ is dense in $\mathbb{T}$ for
every $k$. Using a compactness argument, we find finitely many numbers $n_{j}$
and $k_{j}$ such that $T^{n_{j}}(U_{k_{j}})\subseteq U_{k}$ for every $j$ and
such that

(3) $\mathbb{T}\subseteq\bigcup_{j=1}^{N}B(\lambda^{n_{j}},\varepsilon).$

We deduce that there is $l$ such that $T^{n_{j}}(U_{l})\subseteq U_{k}$ for
every $j$. Let $x\in U_{l}$ such that $A_{l}:=N_{T}(x,U_{l})\in\mathcal{F}$.
By (3), the sets $I_{j}=\\{n:\lambda^{n+n_{j}}\in B(1,\varepsilon)\\}$ form a
cover of $\mathbb{N}$. If $n\in A_{l}\cap I_{j}$ we have that

$\displaystyle\lambda^{n+n_{j}}T^{n+n_{j}}(x)\in
B(1,\varepsilon)T^{n_{j}}(U_{l})\subseteq B(1,\varepsilon)U_{k}\subseteq U.$

It follows by the CuSP property that $N_{\lambda T}(x,U)\in\mathcal{F}$.

The case $\Lambda=G_{n}$ is similar. ∎

It is known that the family of sets with lower or positive density, of
positive upper Banach density, the family of syndetic sets and the family of
infinite sets have the CuSP property (see [6, Lemma 4.1]), while the families
$\mathcal{AP}_{b}$ and $\mathcal{IAP}$ don’t (see [8, Remark 3.23]). We will
use the following corollary in the next theorem.

######  Corollary 3.3.

If $T$ has the $\mathscr{P}_{\overline{\mathcal{BD}}}$ property then for any
$|\lambda|=1$, $\lambda T$ has the $\mathscr{P}_{\overline{\mathcal{BD}}}$
property.

######  Theorem 3.4.

Let $T$ be a linear operator such that $T$ satisfies
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property and let
$\lambda\in\mathbb{T}$. Then either $T=\lambda Id$ or $T-\lambda Id$ is not
quasinilpotent.

###### Proof.

Suppose that $T\neq\lambda Id$. We have to prove that $T-\lambda Id$ is not
quasinilpotent. By Corollary 3.3 we may assume that $\lambda=1$. We will prove
that $T^{**}-Id$ is not quasinilpotent.

Since $T\neq Id$, there is $x\in X$ such that $x\neq T(x)$ and hence there is
$x^{*}\in X^{*}$ such that $Re(\left\langle x,x^{*}\right\rangle)>0$ and
$Re(\left\langle T^{**}(x),x^{*}\right\rangle)<0$.

Therefore if we consider $U:=\\{y\in X^{**}:Re(\left\langle
y,x^{*}\right\rangle)>0,Re(\left\langle T^{**}(y),x^{*}\right\rangle)<0\\}$ we
have that $U\cap X\neq\emptyset$. There is, by Theorem 2.13, $x^{**}\in
X^{**}$ such that $N_{T^{**}}(x^{**},U)$ has positive lower density. It
follows by Lemma 3.1 that $T^{**}-Id$ is not quasinilpotent. ∎

As an immediate corollary we see that operators having the
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property have perfect spectrum.

######  Corollary 3.5.

If $T$ has the $\mathscr{P}_{\overline{\mathcal{BD}}}$ property then either
$T$ has perfect spectrum or $T$ is an unimodular multiple of the identity.

###### Proof.

Suppose that $T$ has isolated spectrum. By Riesz decomposition Theorem we may
assume that $\sigma(T)=\\{\lambda\\}$. Since $T$ is recurrent, it follows that
$\lambda\in\mathbb{T}$ (see [10]). Hence $T-\lambda Id$ is quasinilpotent
which implies that $T=\lambda Id$. ∎

######  Corollary 3.6.

Let $X$ be a Fréchet space, and $T=\lambda Id+K$, with $K$ compact. If $T$ has
the $\mathscr{P}_{\overline{\mathcal{BD}}}$ property then $|\lambda|=1$ and
$K=0$.

###### Proof.

If $X$ is a Banach space, then it is immadiate from Corollary 3.5. If $X$ is a
Fréchet space, the result follows from the existence of a Banach space and a
linear operator $S$ on it, which is a compact perturbation of $\lambda Id$ and
which is conjugate to $T$ (see for example, [21, Proposition 6.1]). ∎

It is known that there are Banach spaces such that every linear operator is a
compact perturbation of a multiple of the identity [1]. Since any such
operator has discrete spectrum, we have the following.

######  Corollary 3.7.

There are infinite dimensional separable Banach spaces such that every
operator with the $\mathscr{P}_{\overline{\mathcal{BD}}}$ property is a
unimodular multiple of the identity.

In particular, there are infinite dimensional separable Banach spaces without
reiteratively hypercyclic operators.

##  4\.  Backward shifts and counterexamples for non $w^{*}$-$w^{*}$
continuous operators

In this section we characterize backward shift operators having the
$\mathscr{P}_{\mathcal{F}}$ property. We also give counterexamples of Theorem
2.10 for non $w^{*}$-$w^{*}$ continuous operators.

It was proved in [6] that if there is a dense set of vectors satisfying
$T^{n}(x)\to 0$, then any $\mathcal{F}$-recurrent operator is also
$\mathcal{F}$-hypercyclic. The same proof holds for operators having the
$\mathscr{P}_{\mathcal{F}}$ property.

######  Proposition 4.1.

Let $\mathcal{F}$ be an upper family and let $T$ be a linear operator such
that there is a dense set of vectors with $T^{n}(x)\to 0$. If $T$ has the
$\mathscr{P}_{\mathcal{F}}$ property it is $\mathcal{F}$-hypercyclic.

###### Proof.

Let $U,V$ be nonempty open sets, and take $W_{0},W$ nonempty open sets such
that $0\in W_{0}$ and $W+W_{0}\subseteq V$. Hence, there is $y\in W$ and
$\delta$ such that $N(y,W)\in\mathcal{F}_{\delta}$. $x\in U-y$ such that
$T^{n}(x)\to 0$. Therefore $x+y\in U$ and for large enough $n$, $T^{n}(x)\in
W_{0}$. Hence if $n\in N(y,W)$ is large enough, $T^{n}(x+y)\in
W+W_{0}\subseteq V$. Therefore, $N(x+y,V)\in\mathcal{F}_{\delta}$. By Theorem
1.1 this implies that $T$ is $\mathcal{F}$-hypercyclic. ∎

######  Corollary 4.2.

Let $T$ be a $w^{*}$-$w^{*}$ continuous operator with a dense set of vectors
such that $T^{n}(x)\to 0$ and let $\mathcal{F}$ be an upper family. Then $T$
has the $\mathscr{P}_{b\mathcal{F}}$ property if and only if it is
$\mathcal{F}$-hypercyclic.

###### Proof.

If $T$ has the $\mathscr{P}_{b\mathcal{F}}$ then it has, by Theorem 2.10, the
$\mathscr{P}_{\mathcal{F}}$. By the above proposition it is
$\mathcal{F}$-hypercyclic. ∎

We mention that Proposition 4.1 and Corollary 4.2 are no longer true for
general (not necessarily upper) families, see Remark 4.5.

In [4] it was proved that for $\ell_{p}$ spaces, reiteratively hypercyclic
backward shifts are indeed frequently hypercyclic and the main Theorem of [9]
states that upper frequently backward shifts along a boundedly complete
unconditional basis are both frequently hypercyclic and chaotic. Recall that a
basis $\\{e_{n}\\}$ is boundedly complete if the series $\sum_{k}a_{k}e_{k}$
is convergent whenever the sequence of partial sums $\Big{(}\sum_{k\leq
n}a_{k}e_{k}\Big{)}_{n}$ is bounded. The following theorem extends both
results.

######  Theorem 4.3.

Let $X$ be a Banach space with boundedly complete bounded basis
$\\{e_{n}:\,n\in\mathbb{N}\\}$ and suppose that $B(e_{n})=e_{n-1}$ is
continuous. The following are equivalent.

  1. i)

$B$ has the $\mathscr{P}_{\overline{\mathcal{BD}}}$ property;

  2. ii)

$B$ is reiteratively hypercyclic;

  3. iii)

$B$ is upper frequently hypercyclic.

  4. iv)

$B$ is $\overline{\mathcal{LD}}$\- hypercyclic.

If moreover, the basis is unconditional, then i)-iv) are equivalent to

  1. v)

$B$ is frequently hypercyclic and

  2. vi)

$B$ is chaotic.

###### Proof.

Since
$b\overline{\mathcal{LD}}=b\overline{\mathcal{D}}=\overline{\mathcal{BD}}$, by
Corollary 4.2, it suffices to show that $B$ is $w^{*}$-$w^{*}$ continuous. It
is well known that $\\{e_{n}\\}$ is boundedly complete if and only if $X$ is
isomorphic to $H^{*}$, where $H=\overline{span\\{e_{n}^{*}\\}}\subseteq
X^{*}$. The isomorphism is given by the mapping $j:X\to H^{*}$, defined as
$j(x)(h)=h(x)$.

Consider $S:=B^{*}:X^{*}\to X^{*}$, or equivalently $B^{*}:H^{**}\to H^{**}$.
Now we notice that $S(e_{n}^{*})=e_{n+1}^{*}$. This implies that
$S(H)=S(\overline{span\\{e_{n}^{*}\\}})\subseteq\overline{span\\{e_{n}^{*}\\}}=H$,
i.e. $S|_{H}:H\to H$ is well defined and hence $(S|_{H})^{*}=B$. Therefore,
$B$ is $w^{*}$-$w^{*}$ continuous.

The last assertion now follows from [9, Theorem 2.1]. ∎

Recall that a Furstenberg family $\mathcal{F}$ is said to be finitely
invariant if $A\cap[m,\infty)\in\mathcal{F}$ for every $A\in\mathcal{F}$ and
every $m\in\mathbb{N}$. Bonilla and Grosse-Erdmann characterized
$\mathcal{F}$-hypercyclic backward shifts for upper finitely invariant
families $\mathcal{F}$, see [5, Theorem 5.1]. We will show next that the
conditions in [5, Theorem 5.1] are actually the conditions that characterize
backward shifts with $\mathscr{P}_{\mathcal{F}}$ property for (non necessarily
upper) finitely invariant families. When the family is also upper, we know by
Proposition 4.1 that $\mathscr{P}_{\mathcal{F}}$ property implies
$\mathcal{F}$-hypercyclicity.

######  Theorem 4.4.

Let $X$ be Banach space with basis $\\{e_{n}:\,n\in\mathbb{N}\\}$ and suppose
that the backward shift is well defined. Let $\mathcal{F}$ be a finitely
invariant Furstenberg family. Consider the following statements.

  1. i)

$B$ has the $\mathscr{P}_{\mathcal{F}}$ property;

  2. ii)

for every $\varepsilon>0$ and every $p$ there is a set
$A_{p,\varepsilon}\in\mathcal{F}$ such that

    1. (a)

$\sum_{n\in A_{p,\varepsilon}}e_{n+p}$ converges and

    2. (b)

for any $m\in A_{p,\varepsilon}$ and any $0\leq j\leq p$,

$\Big{\|}\sum_{n>m,n\in A_{p,\varepsilon}}e_{n-m+j}\Big{\|}<\varepsilon.$

Then $ii)\Rightarrow i)$. If the basis is unconditional $ii)\Leftrightarrow
i)$.

###### Proof.

By the continuity of $B$ there is $\beta_{p}$ such that $\|x\|<\beta_{p}$ then
$\max\\{|x_{j}|,\|B^{j}(x)\|\\}<1/2$ for every $j\leq 2p$.

$i)\Rightarrow ii).$ Let $p,\varepsilon>0.$ Let $y=\sum_{j=1}^{p}je_{j}$ and
$x$ such that
$A_{p,\varepsilon}:=\\{n:\|B^{n}(x)-y\|<min\\{\frac{\varepsilon}{2C},\beta_{p}\\}\\}\in\mathcal{F}$,
where $C$ is the unconditional basis constant.

We claim that $A_{p,\varepsilon}$ satisfies conditions $(a)$ and $(b)$.

We notice first that if $n\in A_{p,\varepsilon}$ and $1\leq j\leq p$, then
$|x_{n+j}-j|<\frac{1}{2}$ and then $\frac{1}{|x_{n+j}|}<2$. Since the basis is
unconditional and $x\in X$ we obtain that

$\sum_{n\in A_{p,\varepsilon}}e_{n+p}=\sum_{n\in
A_{p,\varepsilon}}\frac{1}{x_{n+p}}x_{n+p}e_{n+p}$

converges in $X$. This shows condition $(a)$.

Let $m\in A_{p,\varepsilon}$. Then, for $1\leq j\leq p$,

$\left\|\sum_{n\in
A_{p,\varepsilon},n>m}\frac{1}{x_{n+j}}[B^{m}(x)-y]_{n-m+j}e_{n-m+j}\right\|\leq
2C\|B^{m}(x)-y\|<\varepsilon.$

We observe now that if $m<n\in A_{p,\varepsilon}$ and $j,k\leq p$ then
$|[B^{n}(x)-y]_{j}-[B^{m}(x)-y]_{k}|=|x_{n+j}-x_{m+k}-j+k|<1$. Since $n>m$,
this implies that $n+j\neq m+k$ for every $j,k\leq p$ and thus $n>m+p$.
Therefore $[B^{m}x-y]_{n-m+j}=x_{n+j}$ and thus for $1\leq j\leq p$,

$\|\sum_{n>m}e_{n-m+j}\|<\varepsilon.$

Finally, the case $j=0$ follows from the case $j=1$ and the continuity of $B.$

$ii)\Rightarrow i).$ By the finite invariance of $\mathcal{F}$, we may replace
condition $(a)$ by

$(a^{\prime})$ for any $p$,

$\|\sum_{n\in A_{p,\varepsilon}}e_{n+p}\|<\beta_{p}.$

We may assume also that $n-m>p$ for any $n>m\in A_{p,\varepsilon}$ . Indeed,
by considering $\varepsilon<\beta_{p}$, then $\|\sum_{n>m,n\in
A_{p,\varepsilon}}e_{n-m}\|<\beta_{p}$. Then, by the definition of
$\beta_{p}$, for every $j\leq 2p$, the $j$-th coordinate of $\sum_{n>m,n\in
A_{p,\varepsilon}}e_{n-m}$ is smaller than $\frac{1}{2}$, and this implies
that $n-m>p$.

Let $y=\sum_{j=1}^{p}y_{j}e_{j}\in X$ and $\varepsilon>0.$ Let
$A_{p,\varepsilon}\in\mathcal{F}$ satisfying conditions $(a^{\prime})$ and
$(b)$. We define $x=\sum_{n\in A_{p,\varepsilon}}\sum_{j=1}^{p}y_{j}e_{n+j}$.
The vector is well defined, because

$\displaystyle\|\sum_{n\in A_{p,\varepsilon}}\sum_{j=1}^{p}y_{j}e_{n+j}\|$
$\displaystyle=\|\sum_{j=1}^{p}y_{j}B^{p-j}(\sum_{n\in
A_{p,\varepsilon}}e_{n+p})\|\leq\sum_{j=1}^{p}|y_{j}|\|B^{p-j}\sum_{n\in
A_{p,\varepsilon}}e_{n+p}\|\leq\frac{1}{2}\sum_{j=1}^{p}|y_{j}|<\infty.$

Notice that if $m,n\in A_{p,\varepsilon}$, $m>n$ and $j\leq p$ then $n-m+j\leq
0$. Consequently $B^{m}(e_{n+j})=0$. This implies that if $m\in
A_{p,\varepsilon}$

$\displaystyle B^{m}(x)-y=\sum_{n\in
A_{p,\varepsilon},n>m}\sum_{j=1}^{p}y_{j}e_{n-m+j}=\sum_{j=1}^{p}y_{j}\sum_{n>m,n\in
A}e_{n+j-m}$

and thus, by $(b),$

$\|B^{m}(x)-y\|\leq\sum_{j=1}^{p}|y_{j}|\Big{\|}\sum_{n\in
A_{p,\varepsilon},n>m}e_{n-m+j}\Big{\|}<\varepsilon\sum_{j=1}^{p}|y_{j}|.$

Therefore $B$ has the $\mathscr{P}_{\mathcal{F}}$ property.

∎

######  Remark 4.5.

As mentioned above, in [5, Theorem 5.1] it was proved that for upper finitely
invariant families, the conditions in Theorem 4.4 characterize when a
unilateral backward shift is $\mathcal{F}$-hypercyclic. This follows also from
the above Theorem 4.4 and Proposition 4.1. However these conditions do not
characterize $\mathcal{F}$-hypercyclic backward shifts for general (non upper)
finitely invariant families. Indeed, any chaotic operator has the
$\mathscr{P}_{\mathcal{S}yn}$ property. In particular there exist backward
shift operators with the $\mathscr{P}_{\mathcal{S}yn}$ property (which are
characterized by Theorem 4.4). But, on the other hand, there are no
${\mathcal{S}yn}$-hypercyclic operators, see [4, Proposition 3] (or Corollary
2.15).

On the other hand, it is unknown whether any backward shift with the
$\mathscr{P}_{\underline{\mathcal{D}}}$ property is automatically frequently
hypercyclic.

######  Corollary 4.6.

Let $X$ be a Banach space with unconditional basis and suppose that the
backward shift operator $B$ has the $\mathscr{P}_{\mathcal{S}yn}$ property.
Then $B$ is chaotic.

###### Proof.

Recall that $B$ is chaotic if and only if $\sum_{n}e_{n}$ converges. If $B$
has the $\mathscr{P}_{\mathcal{S}yn}$ property then, by Theorem 4.4, there is
a syndetic set $A$ such that $\sum_{n\in A}e_{n}$ converges. Since $A$ has
bounded gaps, there is $b>0$ such that $\mathbb{N}\subseteq\bigcup_{j\leq
b}A-j$.

If $C>0$ denotes the constant of the basis we obtain that

$\|\sum_{n\in\mathbb{N}}e_{n}\|\leq C\|\sum_{j\leq b}B^{j}\left(\sum_{n\in
A}e_{n}\right)\|.$

∎

######  Example 4.7.

In [4] an example of a non upper frequently hypercyclic but reiteratively
hypercyclic backward shift over $c_{0}$ was given. Since any unilateral
backward shift has a dense set of vectors such that $B^{n}(x)\to 0$,
Proposition 4.1 implies that this operator does not have the
$\mathscr{P}_{\overline{\mathcal{D}}}$ property. This shows in particular an
example of a non $w^{*}$-$w^{*}$ continuous operator such that $T$ has the
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property but not the
$\mathscr{P}_{\overline{\mathcal{D}}}$ property.

In [8] it was proved that the same operator is also
$\mathcal{AP}\\!_{b}$-hypercyclic. Hence it is $\mathcal{PS}yn$-hypercyclic.
We notice that the operator is not chaotic, because chaotic backward shifts
are frequently hypercyclic. Thus, by the above corollary, the operator does
not have the $\mathscr{P}_{\mathcal{S}yn}$ property. This shows in particular,
that the condition on $w^{*}$-$w^{*}$-continuity in Theorem 2.10 and Corollary
2.12 cannot be dropped.

In [3, Theorem 5] (see also [5]) it is shown an example of an upper frequently
backward shift that is not frequently hypercyclic. Actually their proof shows
also that this is an example of an operator with the
$\mathscr{P}_{{\overline{\mathcal{D}}}}$ property which does not satisfy
$\mathscr{P}_{\underline{\mathcal{D}}}$ property.

######  Theorem 4.8.

Let $\mathcal{F}$ be a finitely invariant Furstenberg family. Let
$\\{e_{n}:\,n\in\mathbb{Z}\\}$ be a basis of the Banach space $X$ and $B$ a
well defined backward shift. Consider the following statements.

  1. i)

$B$ has the $\mathscr{P}_{\mathcal{F}}$ property;

  2. ii)

for every $\varepsilon>0$ and every $p$ there is a set
$A=A_{p,\varepsilon}\in\mathcal{F}$ such that

    1. (a)

$\sum_{n\in A}e_{n+p}$ converges and,

    2. (b)

for any $m\in A$ and any $|j|\leq p$,

$\Big{\|}\sum_{n\neq m,n\in A}e_{n-m+j}\Big{\|}<\varepsilon.$

Then $ii)\Rightarrow i)$. If the basis is unconditional $ii)\Leftrightarrow
i)$.

We will only sketch its proof because it is similar to the proof Theorem 4.4.

###### Proof.

By the continuity of $B^{k}$ there is $\beta_{p}$ such that $\|x\|<\beta_{p}$
then $\max\\{|x_{j}|,\|B^{j}(x)\|\\}<1/2$ for every $j\leq 2p$.

$i)\Rightarrow ii)$ Let $y=\sum_{-p\leq j\leq p}je_{j}$. Thus, there are $x\in
X$ and $A\in\mathcal{F}$ such that
$\|B^{n}(x)-y\|<\min\\{\frac{\varepsilon}{C},\beta_{p}\\}$ for every $n\in A$,
where $C$ is the unconditional basis constant. Then $A$ will satisfy
conditions $(a)$ and $(b)$.

For $n\in A$, $|j|\leq p$, $|x_{n+j}-j|<\frac{1}{2}$ and thus
$|\frac{1}{x_{n+j}}|<2$ for $j\neq 0$. Unconditionality then implies that
$\sum_{n\in A}e_{n+p}=\sum_{n\in A}\frac{1}{x_{n+p}}x_{n+p}e_{n+p}$ converges
and thus we have $(a)$.

For $(b)$, let $m\in A$. Then we see as in the proof of Theorem 4.4 that
$[B^{m}(x)-y]_{n-m+j}=x_{n+j}$ for every $n\in A$ and $|j|\leq p$. Then, for
$|j|\leq p$, $j\neq 0$,

$\displaystyle\Big{\|}\sum_{n\in A,n\neq
m}e_{n-m+j}\Big{\|}=\Big{\|}\sum_{n\in A,n\neq
m}\frac{1}{x_{n+j}}[B^{m}(x)-y]_{n-m+j}e_{n-m+j}\Big{\|}\leq
2C\|B^{m}(x)-y\|\leq\varepsilon.$

$ii)\Rightarrow i)$ We may assume that $A$ satisfies $(a^{\prime})$: for any
$p,$ $\|\sum_{n\in A}e_{n+p}\|<\beta_{p}.$

Also, considering $\varepsilon<\beta_{p}$, we may also assume that $|n-m|>p$
for any $n\neq m\in A$.

Let $y=\sum_{j=-p}^{p}y_{j}e_{j}$ and $A\in\mathcal{F}$ satisfying conditions
$(a^{\prime})$ and $(b)$ and $x=\sum_{n\in A}\sum_{-p}^{p}y_{j}e_{n+j}$. Since
$x=\sum_{j=-p}^{p}y_{j}B^{p-j}(\sum_{n\in A}e_{n+p})$, it is well defined
$(a^{\prime})$. And, by $(b)$, if $m\in A$ we have that

$\displaystyle\Big{\|}B^{m}(x)-y\Big{\|}=\Big{\|}\sum_{n\neq m,n\in
A}\sum_{j=-p}^{p}y_{j}e_{n+j-m}\Big{\|}=\Big{\|}\sum_{j=-p}^{p}y_{j}\sum_{n\neq
m,n\in A}e_{n+j-m}\Big{\|}<\varepsilon\sum_{j=-p}^{p}|y_{j}|.$

Therefore $B$ has the $\mathscr{P}_{\mathcal{F}}$ property.

∎

##  5\. Final comments

While we were writing the manuscript we learned about the article [15]. There,
it is proved that adjoint operators that have the
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property support an invariant measure,
thus being frequently recurrent. From this result, part (1) of Corollary 2.12
can immediately be deduced. On the other hand, it is noteworthy to mention
that an application of Theorem 2.10 together with the known results on linear
dynamics from [16] provides an alternative proof of their result.

######  Theorem 5.1 (Grivaux and López-Martínez [15]).

Let $T$ be an adjoint operator that has the
$\mathscr{P}_{\overline{\mathcal{BD}}}$ property. Then $T$ supports an
invariant measure with full support.

###### Proof.

Since $T$ has the $\mathscr{P}_{\overline{\mathcal{BD}}}$ property and it is
$w^{*}$-$w^{*}$ continuous, by Theorem 2.10 , it has the
$\mathscr{P}_{\mathcal{D}}$ property. Theorem [16, Proposition 2.11] then
implies that $T$ supports an invariant measure with full support. ∎

We would like to end the article with some open problems on the
$\mathscr{P}_{\mathcal{F}}$ property.

It is clear that an operator with a dense set of $\mathcal{F}$-recurrent
vectors has the $\mathscr{P}_{\mathcal{F}}$ property. If
$\mathcal{F}=\mathcal{F}_{\infty}$, the family of infinite subsets, then
$\mathcal{F}_{\infty}$-recurrence is just recurrence and by [10, Proposition
2.1], the $\mathscr{P}_{\mathcal{F}_{\infty}}$ property implies
$\mathcal{F}_{\infty}$-recurrence. The implication $\mathscr{P}_{\mathcal{F}}$
property $\Rightarrow\,\mathcal{F}$-recurrence also holds when
$\mathcal{F}=\mathcal{AP}$ is the family of subsets containing arbitrarily
long arithmetic progressions (see [7, Proposition 5.4]). If the operator is an
adjoint operator then we know of some other instances where this implication
holds: when $\mathcal{F}=\mathcal{AP}\\!_{b}$ or $\mathcal{F}=\mathcal{IAP}$
(see Corollary 2.12), when $\mathcal{F}=\overline{\mathcal{BD}},$
$\mathcal{F}=\overline{\mathcal{D}},$ or $\mathcal{F}=\underline{\mathcal{D}}$
(see [15]). We don’t know the following.

######  Question 5.2.

Let $\mathcal{F}$ be a Furstenberg family, does $\mathscr{P}_{\mathcal{F}}$
property imply $\mathcal{F}$-recurrence?

A related problem is whether the analogous of Theorem 2.13 holds for
$\mathcal{F}$-recurrence.

######  Question 5.3.

Let $T$ be a $w^{*}$-$w^{*}$ continuous operator that is
$b\mathcal{F}$-recurrent. Is $T$ $\mathcal{F}$-recurrent?

Note that a positive answer to Question A implies a positive answer to
Question B.

By Theorem 4.4, any unilateral backward shift operator is
$\mathcal{F}$-hypercyclic whenever it has the $\mathscr{P}_{\mathcal{F}}$
property and $\mathcal{F}$ is an upper family. The analogous Theorem is
unclear for bilateral backward shifts.

######  Question 5.4.

Let $\mathcal{F}$ be an upper family and let $T$ be a bilateral backward shift
having the $\mathscr{P}_{\mathcal{F}}$ property. Is $T$
$\mathcal{F}$-hypercyclic?

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