# Common hypercyclic algebras for families of products of backward shifts

Fernando Costa Jr  Université d’Avignon et des Pays de Vaucluse, Laboratoire
de Mathématiques, Campus Jean-Henri Fabre, 301, rue Baruch de Spinoza, BP
21239, 84 916 Avignon Cedex 9 France

###### Abstract.

In this paper, we generalize to the context of algebras some recent results on
the existence of common hypercyclic vectors for families of products of
backward shift operators. We also give, in a multi-dimensional setting, a
positive answer to a question raised by F. Bayart, D. Papathanasiou and the
author about the existence of a common hypercyclic algebra on
$\ell_{1}(\mathbb{N})$ with the convolution product for the family of backward
shifts $(B_{w(\lambda)})_{\lambda>0}$ induced by the weights
$w_{n}(\lambda)=1+\lambda/n$.

###### Key words and phrases:

Common hypercyclicity, weighted shifts, hypercyclic algebras

###### 1991 Mathematics Subject Classification:

47A16

The author was partially supported by the grant ANR-17-CE40-0021 of the French
National Research Agency ANR (project Front)
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##  1\. Introduction

The study of hypercyclicity consists of analysing the behaviour of the
iterates of an operator $T$ on a vector $x$ in a topological vector space $X$.
The set of iterates is called the orbit of $x$ under $T$ and denoted by
$Orb(T;x):=\\{T^{n}(x):n\geq 1\\}$. When the latter is dense in $X$, we say
that $T$ is a _hypercyclic operator_ and that $x$ is one of its _hypercyclic
vectors_. The set of hypercyclic vectors of $T$ is denoted by $HC(T)$ (see [7]
or [9] to learn more on the general theory of linear dynamical systems). When
we are dealing with a family $(T_{\lambda})_{\lambda\in\Lambda}$ of
hypercyclic operators acting on the same space $X$, it is natural to ask
whether there is a single vector $x\in X$ which is hypercyclic for each member
$T_{\lambda}$ of the family. We usually assume that the parameter set
$\Lambda$ is $\sigma$-compact and that $(\lambda,x)\mapsto T_{\lambda}(x)$ is
continuous from $\Lambda\times X$ into $X$. Such a vector is called _common
hypercyclic vector_ for the continuous family
$(T_{\lambda})_{\lambda\in\Lambda}$, which in turn is called _common
hypercyclic family_ of operators.

This question is interesting and non-trivial. On the one hand, many classical
family of operators are common hypercyclic, for example $(\lambda
B)_{\lambda>1}$ on $\ell_{p}(\mathbb{N})$, $p\in[1,+\infty)$, where $B$ is the
backward shift, or $(T_{a})_{a\neq 0}$ on $H(\mathbb{C})$, where
$T_{a}:f\mapsto f(\cdot+a)$ is the operator of translation by $a\neq 0$. On
the other hand, an example due to A. Borichev (quoted in [1]) show that the
family $(sB\times tB)_{(s,t)\in\Lambda}$ does not have common hypercyclic
vectors on $\ell_{2}(\mathbb{N})\times\ell_{2}(\mathbb{N})$ whenever $\Lambda$
has positive Lebesgue measure. Of course, the smaller the parameter set is,
the better is the chance of finding a common hypercyclic vector.

When $X$ has a structure of algebra, it is natural to ask whether or not
$HC(T)\cup\\{0\\}$ contains a non-trivial subalgebra of $X$, what we shall
name as _hypercyclic algebra_ for $T$. Although the subject is not new,
appearing with a negative result in [2, 3] and with a positive result in [10,
7], it is continuously catching the attention of many researchers on the field
in the last few years. In this paper, we are exclusively interested in the
case where $X$ is a Fréchet sequence algebra, that is, we will assume that $X$
is a Fréchet subspace of the space of all sequences
$\omega=\mathbb{C}^{\mathbb{N}}$ and on which a well defined product $\cdot$
satisfies, for all $x,y\in X$ and $q\geq 1$,

$\|x\cdot y\|_{q}\leq\|x\|_{q}\times\|y\|_{q},$

where the non-decreasing separating sequence of seminorms
$(\|\cdot\|_{q})_{q\geq 1}$ induces the complete topology of $X$. We are
mainly interested on families of _weighted backward shift_ operators, that is,
maps $B_{w}:X\to X$ defined by

$B_{w}(x_{0},x_{1},x_{2},...)=(w_{1}x_{1},w_{2}x_{2},...),\quad x\in X,$

where the so called _weight sequence_ $w=(w_{n})_{n}$ is a sequence of
positive real numbers. We say that $w$ is _admissible_ when $B_{w}$ is
continuous.

Two classical products are commonly considered on sequence algebras: the
coordinatewise product and the convolution (or Cauchy) product, respectively
defined, for any sequences $(a_{n})_{n\geq 0},(b_{n})_{n\geq 0}$ in $\omega$,
as

$(a_{n})_{n}\cdot(b_{n})_{n}=(a_{n}b_{n})_{n}\quad\quad\text{and}\quad\quad(a_{n})_{n}\cdot(b_{n})_{n}=\Big{(}\sum_{k=0}^{n}a_{k}b_{n-k}\Big{)}_{n}.$

The first one turns $\ell_{p}(\mathbb{N}),p\geq 1,$ and $c_{0}(\mathbb{N})$
into Fréchet sequence algebras and the second one does the same with
$\ell_{1}(\mathbb{N})$. These products also make $H(\mathbb{C})$ a Fréchet
sequence algebra when we endow it with the sequence of seminorms
$(\|\cdot\|_{q})_{q\geq 1}$ given by

$\left\|\sum_{n=0}^{\infty}a_{n}z^{n}\right\|_{q}=\sum_{n=0}^{\infty}|a_{n}|q^{n}.$

A powerful tool to find common hypercyclic vectors is the well known Costakis-
Sambarino criterion (see [8]). Its hypothesis give a clear way of constructing
the partition of the parameter set required to apply the Basic Criterion for
common hypercyclicity [7, Lemma 7.12] (or Theorem 2.1 for a version adapted to
algebras). On higher dimensions, on the other hand, the construction of the
partition is more delicate as there is no trivial way of ordering the sets in
the partition. While in one dimension, one can arrange the partition in a way
that inequalities such as $\lambda<\lambda^{\prime}$ can be used to get rid of
otherwise problematic factors (see the proof of [6, Theorem 3.12]), in two
dimensions one cannot expect to do anything similar with two parameters
$(\lambda,\mu)$ and $(\lambda^{\prime},\mu^{\prime})$. Hence, proximity
conditions on the elements of the partitions become essential. In this paper,
we make use of the ideas in the recent work [4] to obtain similar results for
common hypercyclic algebras for families of product of backward shift
operators. Let us summarize our main findings.

###  1.1. Coordinatewise product

In the same vein as the results from [4], we have obtained a more general
statement which includes both the following corollaries.

######  Corollary 1.1.

Let $d\geq 1$, $X=\ell_{p}(\mathbb{N})$ or $c_{0}(\mathbb{N})$,
$\alpha\in(0,1/d)$, $I\subset(0,+\infty)$ be $\sigma$-compact and
$(w(\lambda))_{\lambda\in I}$ be a continuous family of weights. Assume that,
for all $I_{0}\subset I$ compact, there exist $D_{1},D_{2},D_{3}>0$ and $N\geq
0$ such that, for all $n\geq N$,

  * •

$a\in I_{0}\mapsto\sum_{i=1}^{n}\log\big{(}w_{i}(a)\big{)}$ is
$D_{1}n^{\alpha}$-Lipschitz;

  * •

$\inf_{a\in I_{0}}w_{1}(a)\cdots w_{n}(a)\geq D_{2}\exp(D_{3}n^{\alpha})$.

Then $\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in I^{d}}$ admits a common hypercyclic
algebra for the coordinatewise product.

######  Corollary 1.2.

Let $d\geq 1$, $X=\ell_{p}(\mathbb{N})$ or $c_{0}(\mathbb{N})$,
$I\subset(0,+\infty)$ be $\sigma$-compact and $(w(\lambda))_{\lambda\in I}$ be
a continuous family of weights. Assume that, for all $I_{0}\subset I$ compact,
there exist $D_{1},D_{2},\gamma>0$ and $N\geq 0$ such that, for all $n\geq N$,

  * •

$a\in I_{0}\mapsto\sum_{i=1}^{n}\log(w_{i}(a))$ is $D_{1}\log(n)$-Lipschitz;

  * •

$\inf_{a\in I_{0}}w_{1}(a)\cdots w_{n}(a)\geq D_{2}n^{\gamma}.$

Then $\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in I^{d}}$ admits a common hypercyclic
algebra for the coordinatewise product.

Thus we have found a large class of families of operators parametrized by
subsets of $\mathbb{R}^{d}$ admitting common hypercyclic algebras.
Particularly, for both $w_{n}(a)=1+\frac{a}{n^{1-\alpha}}$, with
$\alpha\in(0,1/d)$, and $w_{n}(a)=1+\frac{a}{n},$ the family
$\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in(0,+\infty)^{d}}$ admits a common
hypercyclic algebra for the coordinatewise product.

As for the characterization [4, Theorem 2.1], the same holds true for the
hypercyclic algebras for the coordinatewise product under natural
modifications. For a particular case of weights
$\big{(}w_{n}(\lambda)\big{)}_{\lambda}$ with the property that the function
$a\mapsto\sum_{k=1}^{n}\log\big{(}w_{k}(a)\big{)}$ is $n^{\alpha}$-Lipschitz,
we obtained the following equivalence.

######  Proposition 1.3.

Let $\alpha\in(0,1]$ and $w_{1}(a)\cdots w_{n}(a)=\exp(an^{\alpha})$ and
$X=\ell_{p}(\mathbb{N})$ or $c_{0}(\mathbb{N})$ endowed with the
coordinatewise product. The following assertions are equivalent:

  1. (a)

$(B_{w(a)}\times B_{w(b)})_{(a,b)\in\Lambda}$ has a common hypercyclic vector
in $X^{2}$;

  2. (b)

$(B_{w(a)}\times B_{w(b)})_{(a,b)\in\Lambda}$ has a common hypercyclic algebra
in $X^{2}$.

###  1.2. Convolution product

As we might expect, the convolution product is more complicated to be dealt
with. With the coordinatewise product, since it satisfies $\big{(}\sum
a_{n}e_{n}\big{)}^{m}=\sum a_{n}^{m}e_{n}$, one has more liberty to play with
the coefficients without worrying about changes on the supports, we can easily
define $m$-roots for vectors and, as a consequence, the adapted conditions in
the criteria for algebras change only for a few new powers in comparison with
the original criteria for vectors. On the flip side, the convolution product
mixes the supports and is hence less predictable. This behavior usually lead
to more complicated criteria for the existence of algebras under this product,
which are in general not analogous to classical results for the existence of
single vectors.

As remarked by its authors, criterion [6, Theorem 3.12] for the existence of
common hypercyclic algebras fails to apply to the family of weights
$w_{n}(\lambda)=1+\frac{\lambda}{n}$, what can be thought as a surprise as
this weight should not be hard to work with. In fact, for both examples given
by the authors, which are $(\lambda B)_{\lambda>1}$ and $(\lambda
D)_{\lambda>0}$, the functions
$\lambda\mapsto\sum_{k=1}^{n}\log\big{(}w_{k}(\lambda)\big{)}$ are
$n$-Lipschitz and the divergence of $\sum 1/n$ is used in the construction of
the partition. For the weights $w_{n}(\lambda)=1+\frac{\lambda}{n}$, however,
the function $\lambda\mapsto\sum_{k=1}^{n}\log\big{(}w_{k}(\lambda)\big{)}$ is
$\log(n)$-Lipschitz, and the series $\sum_{n}\frac{1}{\log(n)}$ diverges much
faster than $\sum 1/n$. Hence, what one can expect to do is to construct a
partition specifically for this last example and to try to use this faster
divergence to avoid using [6, Lemma 3.11]. This method should also work with
the family of weights $w_{n}(\lambda)=(1+\frac{1}{n})^{\lambda}$ or any other
family for which $w_{1}(\lambda)\cdots w_{n}(\lambda)$ behaves like
$n^{\lambda}$.

One could think that the fast divergence of $\sum\frac{1}{\log(n)^{d}}$ could
lead to a positive result in higher dimensions. This is indeed true, as we are
going to prove in this paper, but not by using the dyadic covering discussed
in Section 2.1.1 below. In fact, this covering doesn’t seem to be adapted to
$w_{n}(a)=1+\frac{a}{n}$ on $\ell_{1}(\mathbb{N})^{d}$, as it contains too
many elements. Fortunately, the fact that the family induced by
$w_{n}(a)=1+\frac{a}{n}$ satisfies a $\log(n)$-Lipschitz condition (rather
than $n^{\alpha}$-Lipschitz) will allow us to construct a (trivial) covering
adapted to this case and obtain the following result.

######  Theorem 1.4.

If $w_{1}(a)\cdots w_{n}(a)=n^{a}$ (or if $w_{n}(a)=1+\frac{a}{n}$ with a few
modifications in the proof), then, for all $d\geq 1$, the family
$\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in(0,+\infty)^{d}}$ admits a common
hypercyclic algebra on $\ell_{1}(\mathbb{N})^{d}$ with the convolution
product.

##  2\. Coordinatewise product

In what follows we assume that $X$ is a Fréchet sequence algebra under the
coordinatewise product in which $\textrm{span}(e_{i})$ is dense,
$\Lambda\subset\mathbb{R}^{d}$ is a $\sigma$-compact set of parameters and,
for each $\lambda=(\lambda_{1},...,\lambda_{d})\in\Lambda$, the operator
$T_{\lambda}:X^{d}\to X^{d}$ is defined by
$T_{\lambda}:=B_{w(\lambda_{1})}\times\cdots\times B_{w(\lambda_{d})}$, where
$\big{(}w(x)\big{)}_{x}$ is a continuous family of admissible weights and the
map $(\lambda,v)\mapsto T_{\lambda}(v)$ is continuous. Also, for each
$\lambda\in\Lambda$ and $m\in\mathbb{N}$, we let
$\mathcal{D}:=\prod_{d}\textrm{span}(e_{i})$ and define
$S_{\lambda,m}:\mathcal{D}\to\mathcal{D}$ by

$S_{\lambda,m}:=F_{w(\lambda_{1})^{-1/m}}\times\cdots\times
F_{w(\lambda_{d})^{-1/m}},$

where each $F_{w(x)^{-1/m}}$, $x\in I$, is the weighted forward shift on $X$
with weight sequence $\big{(}w_{n}^{-1/m}(x)\big{)}_{n}$. The symbol
$\|\cdot\|$ represents the $F$-norm of $X$ and thus satisfies, for all $u\in
X$ and $\lambda\in\mathbb{C}$,

  * •

$|\lambda|\leq 1\Rightarrow\|\lambda u\|\leq\|u\|$;

  * •

$\lim_{\lambda\to 0}\|\lambda u\|=0$;

  * •

$\|\lambda u\|\leq(|\lambda|+1)\|u\|.$

We will make use of the following criterion from [6].

######  Theorem 2.1 (Basic Criterion for algebras).

Assume that, for every compact $K\subset\Lambda$, every $m^{\prime}\leq
m^{\prime\prime}$ in $\mathbb{N}$, every $v\in\mathcal{D}$, every $N>0$ and
every $\varepsilon>0$, there exist $q\in\mathbb{N}$,
$\lambda_{0},\dots,\lambda_{q}\in\Lambda$, sets
$\Lambda_{0},\dots,\Lambda_{q}\subset\Lambda$ and positive integers
$n_{0},\dots,n_{q}$ with $n_{0}>N$ and $n_{i+1}-n_{i}>N$ for $i=0,...,q-1$
such that

  1. (I)

$\bigcup_{i}\Lambda_{i}\supset K$;

  2. (II)

for all $\lambda\in\Lambda_{i}$, $i=0,...,q$, and all
$m\in[m^{\prime},m^{\prime\prime}]$,

$\Bigg{\|}\sum_{j}S_{\lambda_{j},m^{\prime}}^{n_{j}}\big{(}v^{1/m^{\prime}}\big{)}\Bigg{\|}\leq\varepsilon\quad\quad\text{and}\quad\quad\Bigg{\|}\sum_{j\neq
i}T_{\lambda}^{n_{i}}\Big{(}S_{\lambda_{j},m^{\prime}}^{n_{j}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}\leq\varepsilon;$

  3. (III)

for all $\lambda\in\Lambda_{i}$, $i=0,...,q$, and all
$m\in(m^{\prime},m^{\prime\prime}]$,

$\Bigg{\|}T_{\lambda}^{n_{i}}\bigg{(}\Big{(}S_{\lambda_{i},m^{\prime}}^{n_{i}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m}\bigg{)}\Bigg{\|}\leq\varepsilon;$

  4. (IV)

for all $\lambda\in\Lambda_{i}$, $i=0,...,q$,

$\bigg{\|}T_{\lambda}^{n_{i}}\Big{(}S_{\lambda_{i},m^{\prime}}^{n_{i}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m^{\prime}}-v\bigg{\|}\leq\varepsilon.$

Then $\bigcap_{\lambda\in\Lambda}HC(T_{\lambda})$ contains an algebra (but
$0$).

###  2.1. A practical criterion for algebras

The following result is a practical criterion which leads to the same
applications as found in [4], but now for hypercyclic algebras.

######  Theorem 2.2.

Let $\alpha\in(0,1/d)$ and let $I\subset(0,+\infty)$ be $\sigma$-compact.
Suppose that, for all $m^{\prime}\in\mathbb{N}$ and all $I_{0}\subset I$
compact, there exist $C_{1},C_{2}>0$, $\beta>\alpha d$,
$F:\mathbb{N}\to(0,+\infty)$ with $F(n)\leq C_{1}n^{\alpha}$ and
$N_{0},M_{0}>0$ such that, for all $n\geq N_{0}$,

  1. (i)

$a\in I_{0}\mapsto\sum_{i=1}^{n}\log(w_{i}(a))$ is $F(n)$-Lipschitz;

  2. (ii)

for all $a\in I_{0}$, $\inf_{a\in I_{0}}w_{1}(a)\cdots w_{k}(a)\to+\infty$ as
$k\to+\infty$;

  3. (iii)

for all $a\in I_{0}$ and $k\geq N_{0}$,

(1)
$\Bigg{\|}\frac{\exp\Big{(}C_{2}\frac{F(n+k)}{(n+k)^{\alpha}}k^{\alpha}\Big{)}}{w_{1}(a)\cdots
w_{k}(a)}e_{k}\Bigg{\|}\leq\frac{M_{0}}{k^{\beta}},$ (2)
$\Bigg{\|}\frac{1}{\big{[}w_{1}(a)\cdots
w_{k}(a)\big{]}^{1/m^{\prime}}}e_{k}\Bigg{\|}\leq\frac{M_{0}}{k^{\beta}}.$

Then $\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in I^{d}}$ admits a common hypercyclic
algebra on $X^{d}$ for the coordinatewise product.

If we assume that $X$ admits a continuous norm, then condition (ii) follows
from (iii) so it can be removed. Indeed, if $X$ has a continuous norm then the
sequence $(e_{n})$ is bounded below.

Before proving this result, we will show that the two practical Corollaries
3.2 and 3.6 from [4] (here stated as Corollaries 1.1 and 1.2) give not only a
common hypercyclic vector but also a common hypercyclic algebra for the
coordinatewise product. The first corollary is proven as follows.

###### Proof of Corollary 1.1.

Let $m^{\prime}\in\mathbb{N}$ and $I_{0}\subset I$ compact. By hypothesis
there are $D_{1},D_{2},D_{3}>0$ and $N\geq 0$ as in the statement. We define
$F(n):=D_{1}n^{\alpha}$, $C_{1}:=D_{1}$ and fix $C_{2}<\frac{D_{3}}{D_{1}}$
and $\beta>\alpha d$. Also we let $N_{0}>N$ big enough so that, for all $k\geq
N_{0}$,

$\min\Big{\\{}D_{2}\exp\big{(}(D_{3}-C_{2}D_{1})k^{\alpha}\big{)},\big{[}D_{2}\exp(D_{3}k^{\alpha})\big{]}^{1/m^{\prime}}\Big{\\}}\geq
k^{\beta}.$

Conditions (i) and (ii) are satisfied by hypothesis and, for all $n,k\geq
N_{0}$ and $a\in I_{0}$, we have

$\displaystyle\Bigg{\|}\frac{\exp\Big{(}C_{2}\frac{F(n+k)}{(n+k)^{\alpha}}k^{\alpha}\Big{)}}{w_{1}(a)\cdots
w_{k}(a)}e_{k}\Bigg{\|}$
$\displaystyle=\frac{\exp\Big{(}C_{2}\frac{D_{1}(n+k)^{\alpha}}{(n+k)^{\alpha}}k^{\alpha}\Big{)}}{w_{1}(a)\cdots
w_{k}(a)}$
$\displaystyle\leq\frac{\exp\Big{(}C_{2}D_{1}k^{\alpha}\Big{)}}{D_{2}\exp\big{(}D_{3}k^{\alpha}\big{)}}$
$\displaystyle=\frac{1}{D_{2}\exp\big{(}(D_{3}-C_{2}D_{1})k^{\alpha}\big{)}}$
$\displaystyle\leq\frac{1}{k^{\beta}}$

and

$\Bigg{\|}\frac{1}{\big{[}w_{1}(a)\cdots
w_{k}(a)\big{]}^{1/m^{\prime}}}e_{k}\Bigg{\|}\leq\frac{1}{\big{[}D_{2}\exp(D_{3}k^{\alpha})\big{]}^{1/m^{\prime}}}\leq\frac{1}{k^{\beta}},$

what verifies (iii) and completes the proof. ∎

######  Example 2.3.

Let $d\geq 1$, $X=\ell_{p}(\mathbb{N})$ or $c_{0}(\mathbb{N})$,
$\alpha\in(0,1/d)$ and $(w(\lambda))_{\lambda>0}$ be the family of weights
defined by $w_{1}(\lambda)\cdots w_{n}(\lambda)=\exp(\lambda n^{\alpha})$ (or
$w_{n}(\lambda)=1+\frac{\lambda}{n^{1-\alpha}}$) for all $n\geq 1$. Then the
family $\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in(0,+\infty)^{d}}$ admits a common
hypercyclic algebra for the coordinatewise product.

The second corollary is proven as follows.

###### Proof of Corollary 1.2.

Let $m^{\prime}\in\mathbb{N}$ and $I_{0}\subset I$ compact. From the
hypothesis there are $D_{1},D_{2},\gamma>0$ and $N\geq 0$ as in the statement.
Fixing $\alpha<\min\big{(}\frac{\gamma}{dm^{\prime}},\frac{1}{d}\big{)}$, one
can find $C_{1}$ big enough so that $D_{1}\log(n)\leq C_{1}n^{\alpha}$ for all
$n\in\mathbb{N}$. Let $\beta\in\big{(}\alpha
d,\frac{\gamma}{m^{\prime}}\big{)}$ and $C_{2}<\frac{\gamma-\beta}{D_{1}}$.
Finally let $N_{0}>N$ big enough so that $n\mapsto\frac{\log(n)}{n^{\alpha}}$
is decreasing on $[N_{0},+\infty)$. It is now easy to apply Theorem 2.2 with
$F(n)=D_{1}\log(n)$. Conditions (i) and (ii) are satisfied by hypothesis and,
for all $a\in I_{0}$ and $n,k\geq N_{0}$,

$\Bigg{\|}\frac{\exp\Big{(}C_{2}\frac{F(n+k)}{(n+k)^{\alpha}}k^{\alpha}\Big{)}}{w_{1}(a)\cdots
w_{k}(a)}e_{k}\Bigg{\|}=\frac{\exp\Big{(}C_{2}D_{1}\frac{\log(n+k)}{(n+k)^{\alpha}}k^{\alpha}\Big{)}}{w_{1}(a)\cdots
w_{k}(a)}\leq\frac{k^{C_{2}D_{1}}}{D_{2}k^{\gamma}}=\frac{D_{2}^{-1}}{k^{\gamma-
C_{2}D_{1}}}\leq\frac{D_{2}^{-1}}{k^{\beta}}$

and

$\Bigg{\|}\frac{1}{\big{[}w_{1}(a)\cdots
w_{k}(a)\big{]}^{1/m^{\prime}}}e_{k}\Bigg{\|}=\frac{1}{\big{[}w_{1}(a)\cdots
w_{k}(a)\big{]}^{1/m^{\prime}}}\leq\frac{1}{\big{[}D_{2}k^{\gamma}\big{]}^{1/m^{\prime}}}\leq\frac{D_{2}^{-1/m^{\prime}}}{k^{\beta}},$

what verifies (iii) and completes the proof. ∎

######  Example 2.4.

Let $d\geq 1$, $X=\ell_{p}(\mathbb{N})$ or $c_{0}(\mathbb{N})$ and
$(w(\lambda))_{\lambda>0}$ be the family of weights defined by
$w_{n}(\lambda)=1+\frac{\lambda}{n}$ (or
$w_{n}(\lambda):=\big{(}1+\frac{1}{n}\big{)}^{\lambda}$). Then the family
$\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in(0,+\infty)^{d}}$ admits a common
hypercyclic algebra for the coordinatewise product.

The problem of finding a common hypercyclic _vector_ in the limit case
$\alpha=1/d$ remains open. Although we don’t intend to explore this
difficulty, it remains as an interesting open problem. We state it here in the
form of a bi-dimensional example.

######  Question 2.5.

Does there exist a set of parameters $\Lambda\subset\mathbb{R}^{2}$ with
positive Lebesgue measure such that the family $(B_{w(\lambda)}\times
B_{w(\mu)})_{(\lambda,\mu)\in\Lambda}$, with
$w_{n}(\lambda)=1+\frac{\lambda}{\sqrt{n}}$, admits a common hypercyclic
vector on $c_{0}(\mathbb{N})\times c_{0}(\mathbb{N})$?

####  2.1.1. A covering lemma

In order to prove that the conditions of Theorem 2.2 are enough to apply the
Basic Criterion, we need a covering result taken from [4]. The $d$-dimensional
version of this lemma holds true but, for simplicity, we will state it for
$d=2$. Let us fix $\alpha\in(0,1/2)$, $\beta>2\alpha$ and $D>0$. An element
$\lambda$ of $\mathbb{R}^{2}$ will be written $\lambda=(x,y)$.

######  Lemma 2.6.

There exists a constant $c>0$ such that, for all $D>0$ and all compact squares
$K\subset\mathbb{R}^{2}$ with $\textrm{diam}(K)\leq cD$, then, for all
$\tau>0$, for all $\eta>0$, for all $N\geq 1$, there exist $q\geq 1$, a
sequence of integers $(n_{j})_{j=0,\dots,q-1}$, a sequence of parameters
$(\lambda_{j})_{j=0,\dots,q-1}$, a sequence of compact squares
$(\Lambda_{j})_{j=0,\dots,q-1}$ such that

  1. (a)

$n_{0}\geq N$, $n_{j+1}-n_{j}\geq N$;

  2. (b)

$K=\bigcup_{j=0}^{q-1}\Lambda_{j}$ and, for all $j=0,\dots,q-1$, writing
$\lambda_{j}=(x_{j},y_{j})$, we have
$\Lambda_{j}\subset\left[x_{j},x_{j}+\frac{\tau}{n_{j}^{\alpha}}\right]\times\left[y_{j},y_{j}+\frac{\tau}{n_{j}^{\alpha}}\right]$;

  3. (c)

for all $0\leq j<l\leq q-1$, for all $\lambda\in\Lambda_{j}$, for all
$\mu\in\Lambda_{l}$,

$\|\lambda-\mu\|_{\infty}\leq\frac{D(n_{l}-n_{j})^{\alpha}}{n_{l}^{\alpha}};$

  4. (d)

$\sum_{j=0}^{q-1}\frac{1}{n_{j}^{\beta}}\leq\eta$;

  5. (e)

for all $j\in\\{0,\dots,q-1\\}$, $\sum_{l\neq
j}\frac{1}{|n_{l}-n_{j}|^{\beta}}\leq\eta$.

This covering works well for families of weights $\big{(}w_{n}(a)\big{)}_{a}$
such that $a\mapsto\sum_{k=1}^{n}\log\big{(}w_{n}(a)\big{)}$ is
$F(n)$-Lipschitz for some function $F:\mathbb{N}\to\mathbb{N}$ satisfying
$F(n)\leq Cn^{\alpha}$, $C>0$, $0<\alpha<1/d$, giving common hypercyclic
vectors for the family of products of $d$ backward shifts. As usual, it is not
very difficult to transfer these results to the context of hypercyclic
algebras for the coordinatewise product, what lead us to the formulation of
Theorem 2.2. As we have discussed, the convolution product is more delicate to
work with.

###### Proof of Theorem 2.2.

For simplicity we prove the result for $d=2$, but it is clear that same can be
done for any $d\in\mathbb{N}$. Also, in order to shrink the notations, we will
write $\widehat{w}_{n}(x)=w_{1}(x)\cdots w_{n}(x)$ for any $x$ and $n$. We aim
to apply the Basic Criterion for algebras with $\Lambda=(0,+\infty)^{d}$ and
some compact square $K\subset I_{0}^{d}$ where $I_{0}\subset(0,+\infty)$ is a
compact interval. Let $m^{\prime}\leq m^{\prime\prime}$ in $\mathbb{N}$ and
$(u,v)\in\mathcal{D}$ with support in $[0,p]$ for some $p\in\mathbb{N}$, say
$u=\sum_{l=0}^{p}u_{l}e_{l}$ and $v=\sum_{l=0}^{p}v_{l}e_{l}$. By hypothesis
we find $C_{1},C_{2}>0$, $\beta>\alpha d$, $F:\mathbb{N}\to(0,+\infty)$ and
$N_{0},M_{0}>0$ as in the statement. Let us fix $D<C_{2}/2^{\alpha}$ and
define

$A_{p}:=\sup\big{\\{}\big{[}\widehat{w}_{l}(x)\|(u,v)\|_{\infty}\big{]}^{m/m^{\prime}}:m\in[m^{\prime},m^{\prime\prime}]\cup\\{1\\},l=0,...,p\text{
and }x\in I_{0}\big{\\}},$
$A_{p}^{\prime}:=\inf\\{\widehat{w}_{l}(x):l=0,...,p\text{ and }x\in
I_{0}\\}.$

We can suppose that $\textrm{diam}(K)\leq cD$, otherwise we subdivide it into
smaller squares. In order to verify the basic criterion, let $N>0$ and
$\varepsilon>0$ arbitrary. Let $\tau_{0}>0$ small enough so that

$\max_{l=0,...,p}\big{(}\|\tau_{0}u_{l}e_{l}\|,\|\tau_{0}v_{l}e_{l}\|\big{)}\leq\frac{\varepsilon}{2(p+1)}$

and let $\tau>0$ small enough so that

$\Big{|}\exp\big{(}C_{1}2^{\alpha}\tau\big{)}-1\Big{|}\leq\frac{\tau_{0}}{A_{p}/A_{p}^{\prime}+\|(u)\|_{\infty}}.$

Finally let us fix $M>\max\\{N_{0},N,p\\}$ big enough so that, for all $n\geq
M$, $l=0,...,p$, $m\in[m^{\prime},m^{\prime\prime}]$ and $a\in I_{0}$, we have
$\widehat{w}_{l+n}(a)\geq 1$ and

$\Bigg{\|}\frac{1}{\big{(}\widehat{w}_{l+n}(a)\big{)}^{1/m^{\prime}}}e_{l}\Bigg{\|}\leq\frac{\varepsilon}{2(p+1)\big{(}(A_{p}/A_{p}^{\prime})\exp(\tau
C_{1}2^{\alpha})+1\big{)}}.$

We then apply Lemma 2.6 for $\tau>0$, for

$\eta=\frac{\varepsilon}{2M_{0}(p+1)(A_{p}+1)}$

and for $M$ and find $q\geq 1$, integers $n_{0},...,n_{q-1}$, parameters
$\lambda_{0},...,\lambda_{q-1}$ and squares $\Lambda_{0},...,\Lambda_{q-1}$
such that properties (a)-(e) hold true. We claim that conditions (I)-(IV) of
Basic Criterion are satisfied.

Condition (I) is automatically true. For condition (II), let
$\lambda\in\Lambda_{i}$ for some $i\in\\{0,...,q\\}$, and let
$m\in[m^{\prime},m^{\prime\prime}]$. We have

$\displaystyle\Bigg{\|}\sum_{j=0}^{q-1}S_{\lambda_{j},m^{\prime}}^{n_{j}}(u,v)\Bigg{\|}$
$\displaystyle=\Bigg{\|}\sum_{j=0}^{q-1}F_{w(x_{j})^{-1/m^{\prime}}}^{n_{j}}(u^{1/m^{\prime}})\Bigg{\|}+\Bigg{\|}\sum_{j=0}^{q-1}F_{w(y_{j})^{-1/m^{\prime}}}^{n_{j}}(v^{1/m^{\prime}})\Bigg{\|}.$

Let us treat each parcel separately. Firstly we apply (2) and get

$\displaystyle\Bigg{\|}\sum_{j=0}^{q-1}F_{w(x_{j})^{-1/m^{\prime}}}^{n_{j}}(u^{1/m^{\prime}})\Bigg{\|}$
$\displaystyle=\Bigg{\|}\sum_{j=0}^{q-1}\sum_{l=0}^{p}\frac{u_{l}^{1/m^{\prime}}}{\big{[}w_{l+1}(x_{j})\cdots
w_{l+n_{j}}(x_{j})\big{]}^{1/m^{\prime}}}e_{l+n_{j}}\Bigg{\|}$
$\displaystyle\leq\sum_{j=0}^{q-1}\sum_{l=0}^{p}\big{(}|\widehat{w}_{l}(x_{j})u_{l}|^{1/m^{\prime}}+1\big{)}\Bigg{\|}\frac{1}{\big{[}\widehat{w}_{l+n_{j}}(x_{j})\big{]}^{1/m^{\prime}}}e_{l+n_{j}}\Bigg{\|}$
$\displaystyle\leq\sum_{j=0}^{q-1}\sum_{l=0}^{p}(A_{p}+1)\frac{M_{0}}{(l+n_{j})^{\beta}}$
$\displaystyle\leq\sum_{j=0}^{q-1}\frac{(p+1)(A_{p}+1)M_{0}}{n_{j}^{\beta}}$
$\displaystyle\leq(p+1)(A_{p}+1)M_{0}\eta$
$\displaystyle\leq\frac{\varepsilon}{2}.$

Analogously we get
$\big{\|}\sum_{j=0}^{q-1}F_{w(y_{j})^{-1/m^{\prime}}}^{n_{j}}(v^{1/m^{\prime}})\big{\|}\leq\frac{\varepsilon}{2},$
what proves the first part of condition (II). For the second, writing
$\lambda=(x,y)$, we have

$\displaystyle\Bigg{\|}\sum_{j\neq
i}T_{\lambda}^{n_{i}}\Big{(}S_{\lambda_{j},m^{\prime}}^{n_{j}}\big{(}(u,v)^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}$
$\displaystyle\leq\Bigg{\|}\sum_{j\neq
i}B_{w(x)}^{n_{i}}\Big{(}F_{w(x_{j})^{-1/m^{\prime}}}^{n_{j}}\big{(}u^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}$
$\displaystyle\quad\quad\quad\quad\quad+\Bigg{\|}\sum_{j\neq
i}B_{w(y)}^{n_{i}}\Big{(}F_{w(y_{j})^{-1/m^{\prime}}}^{n_{j}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}.$

As before, we treat these parcels separately and get

$\displaystyle\Bigg{\|}\sum_{j\neq
i}B_{w(x)}^{n_{i}}\Big{(}F_{w(x_{j})^{-1/m^{\prime}}}^{n_{j}}$
$\displaystyle\big{(}u^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}$
$\displaystyle=\Bigg{\|}\sum_{j>i}B_{w(x)}^{n_{i}}\Big{(}F_{w(x_{j})^{-1/m^{\prime}}}^{n_{j}}\big{(}u^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}$
$\displaystyle=\Bigg{\|}\sum_{j>i}\sum_{l=0}^{p}u_{l}^{m/m^{\prime}}\frac{w_{l+n_{j}-n_{i}+1}(x)\cdots
w_{l+n_{j}}(x)}{\big{[}w_{l+1}(x_{j})\cdots
w_{l+n_{j}}(x_{j})\big{]}^{m/m^{\prime}}}e_{l+n_{j}-n_{i}}\Bigg{\|}$
$\displaystyle\leq\sum_{j>i}\sum_{l=0}^{p}\Bigg{\|}u_{l}^{m/m^{\prime}}\frac{w_{l+n_{j}-n_{i}+1}(x)\cdots
w_{l+n_{j}}(x)}{\big{[}w_{l+1}(x_{j})\cdots
w_{l+n_{j}}(x_{j})\big{]}^{m/m^{\prime}}}e_{l+n_{j}-n_{i}}\Bigg{\|}.$

Notice that, using $m/m^{\prime}\geq 1$ and $D<C_{2}/2^{\alpha}$,

$\displaystyle\Bigg{|}u_{l}^{m/m^{\prime}}\frac{w_{l+n_{j}-n_{i}+1}(x)\cdots
w_{l+n_{j}}(x)}{\big{[}w_{l+1}(x_{j})\cdots
w_{l+n_{j}}(x_{j})\big{]}^{m/m^{\prime}}}\Bigg{|}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq|\widehat{w}_{l}(x_{j})u_{l}|^{m/m^{\prime}}\frac{\widehat{w}_{l+n_{j}}(x)}{\widehat{w}_{l+n_{j}}(x_{j})}\times\frac{1}{\widehat{w}_{l+n_{j}-n_{i}}(x)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq
A_{p}\exp\Bigg{(}\Bigg{|}\sum_{i=1}^{l+n_{j}}\log(w_{i}(x))-\sum_{i=1}^{l+n_{j}}\log(w_{i}(x_{j}))\Bigg{|}\Bigg{)}\times\frac{1}{\widehat{w}_{l+n_{j}-n_{i}}(x)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq
A_{p}\exp\big{(}F(l+n_{j})|x-x_{j}|\big{)}\times\frac{1}{\widehat{w}_{l+n_{j}-n_{i}}(x)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq
A_{p}\exp\Bigg{(}D\frac{F(l+n_{j})}{n_{j}^{\alpha}}(n_{j}-n_{i})^{\alpha}\Bigg{)}\times\frac{1}{\widehat{w}_{l+n_{j}-n_{i}}(x)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq
A_{p}\frac{\exp\Big{(}D\frac{F(l+n_{j})}{(l+n_{j})^{\alpha}}(l+n_{j}-n_{i})^{\alpha}\times\frac{(l+n_{j})^{\alpha}}{n_{j}^{\alpha}}\times\frac{(n_{j}-n_{i})^{\alpha}}{(l+n_{j}-n_{i})^{\alpha}}\Big{)}}{\widehat{w}_{l+n_{j}-n_{i}}(x)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq
A_{p}\frac{\exp\Big{(}D2^{\alpha}\frac{F(l+n_{j})}{(l+n_{j})^{\alpha}}(l+n_{j}-n_{i})^{\alpha}\Big{)}}{\widehat{w}_{l+n_{j}-n_{i}}(x)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq
A_{p}\frac{\exp\Big{(}C_{2}\frac{F(l+n_{j})}{(l+n_{j})^{\alpha}}(l+n_{j}-n_{i})^{\alpha}\Big{)}}{\widehat{w}_{l+n_{j}-n_{i}}(x)}.$

Hence,

$\displaystyle\Bigg{\|}\sum_{j\neq
i}B_{w(x)}^{n_{i}}\Big{(}F_{w(x_{j})^{-1/m^{\prime}}}^{n_{j}}$
$\displaystyle\big{(}u^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}$
$\displaystyle\leq\sum_{j>i}\sum_{l=0}^{p}\Bigg{\|}u_{l}^{m/m^{\prime}}\frac{w_{l+n_{j}-n_{i}+1}(x)\cdots
w_{l+n_{j}}(x)}{\big{[}w_{l+1}(x_{j})\cdots
w_{l+n_{j}}(x_{j})\big{]}^{m/m^{\prime}}}e_{l+n_{j}-n_{i}}\Bigg{\|}$
$\displaystyle\leq\sum_{j>i}\sum_{l=0}^{p}\Bigg{\|}A_{p}\frac{\exp\Big{(}C_{2}\frac{F(l+n_{j})}{(l+n_{j})^{\alpha}}(l+n_{j}-n_{i})^{\alpha}\Big{)}}{\widehat{w}_{l+n_{j}-n_{i}}(x)}e_{l+n_{j}-n_{i}}\Bigg{\|}$
$\displaystyle\leq\sum_{j>i}\sum_{l=0}^{p}\big{(}A_{p}+1\big{)}\Bigg{\|}\frac{\exp\Big{(}C_{2}\frac{F(l+n_{j})}{(l+n_{j})^{\alpha}}(l+n_{j}-n_{i})^{\alpha}\Big{)}}{\widehat{w}_{l+n_{j}-n_{i}}(x)}e_{l+n_{j}-n_{i}}\Bigg{\|}$
$\displaystyle\leq\sum_{j>i}\sum_{l=0}^{p}\big{(}A_{p}+1\big{)}\frac{M_{0}}{(l+n_{j}-n_{i})^{\beta}}$
$\displaystyle\leq
M_{0}(p+1)\big{(}A_{p}+1\big{)}\sum_{j>i}\frac{1}{(n_{j}-n_{i})^{\beta}}$
$\displaystyle\leq M_{0}(p+1)\big{(}A_{p}+1\big{)}\eta$
$\displaystyle\leq\frac{\varepsilon}{2}.$

Analogously, $\Bigg{\|}\sum_{j\neq
i}B_{w(y)}^{n_{i}}\Big{(}F_{w(y_{j})^{-1/m^{\prime}}}^{n_{j}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}\leq\frac{\varepsilon}{2}$,
what shows that condition (II) is satisfied. For condition (III) and (IV), let
$\lambda\in\Lambda_{i}$ for some $i\in\\{0,...,q-1\\}$. We write
$\lambda=(x,y)$. Notice that, for all $l=0,...,p$ and
$(z,z_{i})\in\big{\\{}(x,x_{i}),(y,y_{i})\big{\\}},$

$\displaystyle\frac{\widehat{w}_{l+n_{i}}(z)}{\widehat{w}_{l+n_{i}}(z_{i})}$
$\displaystyle\leq\exp\Bigg{(}\Bigg{|}\sum_{j=1}^{l+n_{i}}\log\big{(}w_{j}(z)\big{)}-\sum_{j=1}^{l+n_{i}}\log\big{(}w_{j}(z_{i})\big{)}\Bigg{|}\Bigg{)}$
$\displaystyle\leq\exp\big{(}F(l+n_{i})|z-z_{i}|\big{)}$
$\displaystyle\leq\exp\Big{(}\tau\frac{F(l+n_{i})}{n_{i}^{\alpha}}\Big{)}$
$\displaystyle\leq\exp\Big{(}\tau
C_{1}\frac{(l+n_{i})^{\alpha}}{n_{i}^{\alpha}}\Big{)}$
$\displaystyle\leq\exp\big{(}\tau C_{1}2^{\alpha}\big{)}.$

Given $m\in(m^{\prime},m^{\prime\prime}]$, we have

$\displaystyle\Bigg{\|}B_{w(x)}^{n_{i}}\Big{(}F_{w(x_{i})^{-1/m^{\prime}}}^{n_{i}}\big{(}u^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}$
$\displaystyle\\!=\\!\Bigg{\|}\sum_{l=0}^{p}u_{l}^{m/m^{\prime}}\frac{w_{l+1}(x)\cdots
w_{l+n_{i}}(x)}{\big{[}w_{l+1}(x_{i})\cdots
w_{l+n_{i}}(x_{i})\big{]}^{\frac{m}{m^{\prime}}}}e_{l}\Bigg{\|}$
$\displaystyle\\!\leq\\!\sum_{l=0}^{p}\Bigg{\|}\frac{\big{(}\widehat{w}_{l}(x_{i})u_{l}\big{)}^{m/m^{\prime}}}{\widehat{w}_{l}(x)}\\!\times\\!\frac{\widehat{w}_{l+n_{i}}(x)}{\widehat{w}_{l+n_{i}}(x_{i})}\\!\times\\!\frac{1}{\big{[}\widehat{w}_{l+n_{i}}(x_{i})\big{]}^{\frac{m}{m^{\prime}}-1}}e_{l}\Bigg{\|}$
$\displaystyle\\!\leq\\!\sum_{l=0}^{p}\Bigg{\|}(A_{p}/A_{p}^{\prime})\exp(\tau
C_{1}2^{\alpha})\frac{1}{\big{[}\widehat{w}_{l+n_{i}}(x_{i})\big{]}^{\frac{m}{m^{\prime}}-1}}e_{l}\Bigg{\|}$
$\displaystyle\\!\leq\\!\big{(}(A_{p}/A_{p}^{\prime})\exp(\tau
C_{1}2^{\alpha})+1\big{)}\sum_{l=0}^{p}\Bigg{\|}\frac{1}{\big{[}\widehat{w}_{l+n_{i}}(x_{i})\big{]}^{1/m^{\prime}}}e_{l}\Bigg{\|}$
$\displaystyle\\!\leq\\!\frac{\varepsilon}{2}.$

Analogously,
$\Bigg{\|}B_{w(y)}^{n_{i}}\Big{(}F_{w(y_{i})^{-1/m^{\prime}}}^{n_{i}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m}\Bigg{\|}\leq\frac{\varepsilon}{2}$,
what verifies condition (III). Finally, for condition (IV), we first notice
that, for any $x,y$ with $|x-y|<\frac{\tau}{n^{\alpha}}$, we have

$\displaystyle\Bigg{|}\\!\Bigg{(}\\!\frac{w_{l+1}(x)\cdots
w_{l+n}(x)}{w_{l+1}(y)\cdots w_{l+n}(y)}-1\\!\\!\Bigg{)}u_{l}\Bigg{|}$
$\displaystyle\\!=\\!\Bigg{|}\\!\Bigg{(}\frac{\widehat{w}_{l}(y)}{\widehat{w}_{l}(x)}\\!\times\\!\frac{\widehat{w}_{l+n}(x)}{\widehat{w}_{l+n}(y)}\\!-\\!\frac{\widehat{w}_{l}(y)}{\widehat{w}_{l}(x)}\\!+\\!\frac{\widehat{w}_{l}(y)}{\widehat{w}_{l}(x)}\\!-\\!1\\!\\!\Bigg{)}u_{l}\Bigg{|}$
$\displaystyle\\!\leq\\!\Bigg{|}\frac{\widehat{w}_{l}(y)}{\widehat{w}_{l}(x)}\Bigg{|}\|u\|_{\infty}\\!\times\\!\Bigg{|}\frac{\widehat{w}_{l+n}(x)}{\widehat{w}_{l+n}(y)}\\!-\\!1\Bigg{|}\\!+\\!\Bigg{|}\frac{\widehat{w}_{l}(y)}{\widehat{w}_{l}(x)}\\!-\\!1\Bigg{|}\|u\|_{\infty}$
$\displaystyle\\!\leq\\!\frac{A_{p}}{A_{p}^{\prime}}\Big{|}\exp\\!\big{(}F(n\\!+\\!l)|x\\!-\\!y|\big{)}\\!-\\!1\Big{|}\\!+\\!\Big{|}\exp\\!\big{(}F(l)|x\\!-\\!y|\big{)}\\!-\\!1\Big{|}\|u\|_{\infty}$
$\displaystyle\\!\leq\\!\frac{A_{p}}{A_{p}^{\prime}}\Big{|}\exp\\!\big{(}F(n+l)\frac{\tau}{n^{\alpha}}\big{)}-1\Big{|}+\Big{|}\exp\\!\big{(}F(l)\frac{\tau}{n^{\alpha}}\big{)}-1\Big{|}\|u\|_{\infty}$
$\displaystyle\\!\leq\\!\frac{A_{p}}{A_{p}^{\prime}}\Big{|}\exp\\!\big{(}C_{1}\tau\frac{(n+p)^{\alpha}}{n^{\alpha}}\big{)}-1\Big{|}+\Big{|}\exp\\!\big{(}C_{1}\tau\frac{p^{\alpha}}{n^{\alpha}}\big{)}-1\Big{|}\|u\|_{\infty}$
$\displaystyle\\!\leq\\!\frac{A_{p}}{A_{p}^{\prime}}\Big{|}\exp\\!\big{(}C_{1}2^{\alpha}\tau\big{)}-1\Big{|}+\Big{|}\exp\\!\big{(}C_{1}\tau\big{)}-1\Big{|}\|u\|_{\infty}$
$\displaystyle\\!\leq\\!(A_{p}/A_{p}^{\prime}+\|u\|_{\infty})\Big{|}\exp\\!\big{(}C_{1}2^{\alpha}\tau\big{)}-1\Big{|}$
$\displaystyle\\!\leq\\!\tau_{0}$

Hence,

$\displaystyle\bigg{\|}T_{\lambda}^{n_{i}}\Big{(}S_{\lambda_{i},m^{\prime}}^{n_{i}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m^{\prime}}-v\bigg{\|}$
$\displaystyle\quad\quad\leq\bigg{\|}B_{w(x)}^{n_{i}}\Big{(}F_{w(x_{i})^{-1/m^{\prime}}}^{n_{i}}\big{(}u^{1/m^{\prime}}\big{)}\Big{)}^{m^{\prime}}-u\bigg{\|}+\bigg{\|}B_{w(y)}^{n_{i}}\Big{(}F_{w(y_{i})^{-1/m^{\prime}}}^{n_{i}}\big{(}v^{1/m^{\prime}}\big{)}\Big{)}^{m^{\prime}}-v\bigg{\|}$
$\displaystyle\quad\quad=\Bigg{\|}\sum_{l=0}^{p}\Bigg{(}\frac{w_{l+1}(x)\cdots
w_{l+n_{i}}(x)}{w_{l+1}(x_{i})\cdots
w_{l+n_{i}}(x_{i})}-1\Bigg{)}u_{l}e_{l}\Bigg{\|}+\Bigg{\|}\sum_{l=0}^{p}\Bigg{(}\frac{w_{l+1}(y)\cdots
w_{l+n_{i}}(y)}{w_{l+1}(y_{i})\cdots
w_{l+n_{i}}(y_{i})}-1\Bigg{)}v_{l}e_{l}\Bigg{\|}$
$\displaystyle\quad\quad\leq\sum_{l=0}^{p}\Big{(}\big{\|}\tau_{0}u_{l}e_{l}\big{\|}+\big{\|}\tau_{0}v_{l}e_{l}\big{\|}\Big{)}$
$\displaystyle\quad\quad\leq\varepsilon.$

This proves the claim and completes the proof. ∎

###  2.2. Characterization of families of products of weighted shifts
admitting a common hypercyclic algebra for the coordinatewise product

As done in [5], many results for vectors can be brought to the context of
algebras with the coordinatewise product under almost the same conditions, the
difference being the presence of some natural powers. Here we repeat this
formula with [4, Theorem 2.1] and obtain Theorem 2.7. For the proof we need
the following property which holds true whenever $(e_{n})_{n}$ is an
unconditional basis of a Fréchet space $X$.

  1. (UB)

If $(x_{n})\in X$ and $(\alpha_{n})\in\ell_{\infty}$, then
$(\alpha_{n}x_{n})\in X$. Moreover, for all $\varepsilon>0$, for all $M>0$,
there exists $\delta>0$ such that, for all $x\in X$ with $\|x\|\leq\delta$,
for all sequence $(\alpha_{n})\in\ell_{\infty}$ with
$\|(\alpha_{n})\|_{\infty}\leq M$, then $(\alpha_{n}x_{n})\in X$ and
$\|(\alpha_{n}x_{n})\|<\varepsilon$.

######  Theorem 2.7.

Let $X$ be a Fréchet sequence algebra for the coordinatewise product with a
continuous norm and admitting $(e_{n})$ as an unconditional basis. Let
$I\subset\mathbb{R}$ be a nonempty interval, let $d\geq 1$ and let
$\Lambda\subset I^{d}$ be $\sigma$-compact. Let $(B_{w(a)})_{a\in I}$ be a
family of weighted shifts on $X$ and assume that $a\in I\mapsto w_{n}(a)$ is
non-decreasing with $\displaystyle\inf_{n\in\mathbb{N},a\in I}w_{n}(a)>0$.
Also, assume that there exist $F:\mathbb{N}\to\mathbb{R}_{+}$ non-decreasing
and $c,C>0$ such that, for all $n\geq 1$, writing
$f_{n}(a)=\sum_{k=1}^{n}\log(w_{k}(a))$,

$\forall(a,b)\in I^{2},\ cF(n)|a-b|\leq|f_{n}(a)-f_{n}(b)|\leq CF(n)|b-a|,$
$\forall(a,b)\in I^{2},\ \frac{w_{n}(a)}{w_{n}(b)}\geq c.$

Then the following assertions are equivalent:

  1. (a)

$(B_{w(\lambda(1))}\times\cdots\times B_{w(\lambda(d))})_{\lambda\in\Lambda}$
admits a common hypercyclic algebra in $X$;

  2. (b)

For all $m\geq 1$, there exist $u\in X^{d}$ such that $u^{m}$ is a common
hypercyclic vector for $(B_{w(\lambda(1))}\times\cdots\times
B_{w(\lambda(d))})_{\lambda\in\Lambda}$;

  3. (c)

For all $m\in\mathbb{N}$, $\tau>0$, for all $N\geq 1$, for all
$\varepsilon>0$, for all $K\subset\Lambda$ compact, there exist $N\leq
n_{1}<n_{1}+N\leq n_{1}<\dots<n_{q-1}+N\leq n_{q}$ and
$(\lambda_{k})_{k=1,\dots,q}\in I^{d}$ such that

    1. (i)

$K\subset\cup_{k=1}^{q}\prod_{l=1}^{d}\left[\lambda_{k}(l)-\frac{\tau}{F(n_{k})},\lambda_{k}(l)\right]$

    2. (ii)

For all $i=1,\dots,d$,

$\left\|\sum_{k=1}^{q}\frac{1}{\big{[}w_{1}(\lambda_{k}(i))\cdots
w_{n_{k}}(\lambda_{k}(i))\big{]}^{1/m}}e_{n_{k}}\right\|<\varepsilon.$

    3. (iii)

For all $k=1,\dots,q$, for all $i=1,\dots,d$, for all $l=0,...,N$,

$\left\|\sum_{j=k+1}^{q}\frac{w_{n_{j}-n_{k}+l+1}(\lambda_{k}(i))\cdots
w_{n_{j}+l}(\lambda_{k}(i))}{w_{l+1}(\lambda_{j}(i))\cdots
w_{n_{j}+l}(\lambda_{j}(i))}e_{n_{j}-n_{k}}\right\|<\varepsilon.$

###### Proof.

To fix ideas, we will prove the case $d=2$. The modifications needed for
obtaining the general case are straightforward, so we let them for the reader.
The notations used are as follows. Each elements of $I^{2}$ will be written as
a couple $\lambda=(a,b)$ and will induce the operators
$T_{\lambda}=B_{w(a)}\times B_{w(b)}$ and its right inverse
$S_{\lambda}=F_{w^{-1}(a)}\times F_{w^{-1}(b)}$ (here, $F_{w^{-1}(a)}$ is the
forward shift induced by $(w_{n}^{-1}(a))_{n}$).

The proof of $(a)\Rightarrow(b)$ is trivial. The proof of $(b)\Rightarrow(c)$
is very similar to what is done in [4], we leave the details to the reader.
Let us show that $(c)\Rightarrow(a)$. We aim to apply the Basic Criterion for
algebras, so let $\mathcal{D}\subset X^{2}$ be the set of couples of vectors
with finite support, let $K\subset\Lambda$ be compact, let $m^{\prime}\leq
m^{\prime\prime}$ and let $v\in\mathcal{D}$ and $N\in\mathbb{N}$, say
$v=(v(1),v(2))$ with $v(i)=\sum_{l=0}^{N}v_{l}(i)e_{l},i=1,2$. Finally we let
$\varepsilon>0$, we fix a small $\tau>0$ (conditions later on) and we apply
(c) with $m=m^{\prime}$ in order to obtain sequences $(n_{k})_{k=1,\dots,q}$
and $(\lambda_{k})_{k=1,\dots,q}$ satisfying (i), (ii) and (iii). We write
$\lambda_{k}=(a_{k},b_{k})$ and set
$\Lambda_{k}=K\cap([a_{k}-\tau/F(n_{k});a_{k}]\times[b_{k}-\tau/F(n_{k});b_{k}])$
so that $\bigcup_{k}\Lambda_{k}\supset K$. We claim that all conditions of the
Basic Criterion for algebras are satisfied for $m_{k}=n_{k}-N$. Condition (I)
is already verified. For later use, we observe that the condition
$\inf_{n}w_{n}(\tilde{a})$ and the unconditionality of $(e_{n})_{n}$ ensures
the continuity of $B_{w(\tilde{a})^{1/m^{\prime}}}$. For condition (II), we
start by verifying that

$\displaystyle\Bigg{\|}\sum_{k=0}^{q}S_{\lambda_{k},m^{\prime}}^{m_{k}}v\Bigg{\|}$
$\displaystyle=\Bigg{\|}\sum_{k=0}^{q}F_{w(a_{k})^{-1/m^{\prime}}}^{m_{k}}(v(1)^{1/m^{\prime}})\Bigg{\|}+\Bigg{\|}\sum_{k=0}^{q}F_{w(b_{k})^{-1/m^{\prime}}}^{m_{k}}(v(2)^{1/m^{\prime}})\Bigg{\|}$

is smaller than $\varepsilon$. For $i=1,2$ we have

$\displaystyle\left\|\sum_{k=1}^{q}F_{w(a_{k})^{-1/m^{\prime}}}^{m_{k}}(v(i)^{1/m^{\prime}})\right\|\leq$
$\displaystyle\quad\quad\leq\sum_{l=0}^{N}(|v_{l}(i)|^{1/m^{\prime}}+1)\left\|\sum_{k=1}^{q}\frac{1}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{1/m^{\prime}}}e_{n_{k}-(N-l)}\right\|.$

Fixing $\tilde{a}\in I$, we see that

(3)
$\displaystyle\begin{split}&\sum_{k=1}^{q}\frac{1}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{1/m^{\prime}}}e_{n_{k}-(N-l)}\\\
&\quad\quad\quad\quad=B_{w(\tilde{a})^{1/m^{\prime}}}^{N-l}\left(\sum_{k=1}^{q}\frac{x_{k,l}}{\big{[}w_{1}(a_{k})\cdots
w_{n_{k}}(a_{k})\big{]}^{1/m^{\prime}}}e_{n_{k}}\right)\end{split}$

for some sequence $(x_{k,l})_{k}\in\ell_{\infty}$ with
$\|(x_{k,l})_{k}\|_{\infty}\leq\left(\frac{M}{c}\right)^{N/m^{\prime}}$, where

$M=\max\big{(}\\{1\\}\cup\\{|w_{j}(a)|:0\leq j\leq N,(a,b)\in K\textrm{ for
some }b\\}\big{)}.$

Effectively,

$\displaystyle\sum_{k=1}^{q}$
$\displaystyle\frac{1}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{1/m^{\prime}}}e_{n_{k}-(N-l)}$
$\displaystyle\quad\quad\quad\quad=\sum_{k=1}^{q}\big{[}w_{1}(a_{k})\cdots
w_{l}(a_{k})\big{]}^{1/m^{\prime}}\times\left[\prod_{j=n_{k}-(N-l)+1}^{n_{k}}\frac{w_{j}(a_{k})}{w_{j}(\tilde{a})}\right]^{1/m^{\prime}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times
B_{w(\tilde{a})^{1/m^{\prime}}}^{N-l}\left(\frac{1}{\big{[}w_{1}(a_{k})\cdots
w_{n_{k}}(a_{k})\big{]}^{1/m^{\prime}}}e_{n_{k}}\right)$
$\displaystyle\quad\quad\quad\quad\leq\sum_{k=1}^{q}M^{N/m^{\prime}}\left(\frac{1}{c}\right)^{(N-l)/m^{\prime}}B_{w(\tilde{a})^{1/m^{\prime}}}^{N-l}\left(\frac{1}{\big{[}w_{1}(a_{k})\cdots
w_{n_{k}}(a_{k})\big{]}^{1/m^{\prime}}}e_{n_{k}}\right).$

Hence, the first condition in (II) follows from the continuity of
$B_{w(\tilde{a})^{1/m^{\prime}}}$ and the unconditionality of $(e_{n})$. For
the second condition in (II), given $\lambda=(a,b)\in\Lambda_{k}$, for some
$k=1,...,q$, and $m\in[m^{\prime},m^{\prime\prime}]$, we have

$\displaystyle\sum_{j\neq k}B_{w(a)}^{m_{k}}$
$\displaystyle\left(F_{w^{-1/m^{\prime}}(a_{j})}^{m_{j}}(v(i)^{1/m^{\prime}})\right)^{m}$
$\displaystyle=\sum_{j=k+1}^{q}B_{w(a)}^{m_{k}}\left(F_{w^{-1/m^{\prime}}(a_{j})}^{m_{j}}(v(i)^{1/m^{\prime}})\right)^{m}$
$\displaystyle=\sum_{j=k+1}^{q}\sum_{l=0}^{N}\big{[}v_{l}(i)\big{]}^{m/m^{\prime}}\frac{w_{n_{j}-n_{k}+l+1}(a)\cdots
w_{n_{j}-(N-l)}(a)}{\big{[}w_{l+1}(a_{j})\cdots
w_{n_{j}-(N-l)}(a_{j})\big{]}^{m/m^{\prime}}}e_{n_{j}-n_{k}+l}$
$\displaystyle=\sum_{j=k+1}^{q}\sum_{l=0}^{N}\big{[}v_{l}(i)\big{]}^{m/m^{\prime}}y_{j,l}\frac{w_{n_{j}-n_{k}+l+1}(a)\cdots
w_{n_{j}-(N-l)}(a)}{w_{l+1}(a_{j})\cdots
w_{n_{j}-(N-l)}(a_{j})}e_{n_{j}-n_{k}+l},$

where

$y_{j,l}=\frac{1}{\big{[}w_{l+1}(a_{j})\cdots
w_{n_{j}-(N-l)}(a_{j})\big{]}^{m/m^{\prime}-1}}.$

Since $X$ has a continuous norm, from (ii) and as $m/m^{\prime}-1\geq 0$ we
get that $(y_{j,l})$ is bounded uniformly on $j$ and $l$. In order to apply
(iii) we define, for $0\leq l\leq N$ and $j\geq k+1$,

$\alpha_{j,l}=y_{j,l}\frac{w_{n_{j}-n_{k}+l+1}(a)\cdots
w_{n_{j}+l}(a)}{w_{n_{j}-n_{k}+l+1}(a_{k})\cdots
w_{n_{j}+l}(a_{k})}\times\frac{w_{n_{j}+l-N+1}(a_{j})\cdots
w_{n_{j}+l}(a_{j})}{w_{n_{j}+l-N+1}(a)\cdots w_{n_{j}+l}(a)}$

and use that

$w_{n_{j}-n_{k}+l+1}(a)\cdots w_{n_{j}+l}(a)\leq
w_{n_{j}-n_{k}+l+1}(a_{k})\cdots w_{n_{j}+l}(a_{k})$

in order to verify that $(\alpha_{j,l})$ is bounded uniformly on $j$ and $l$.
Now,

$\displaystyle\sum_{j\neq k}B_{w(a)}^{m_{k}}$
$\displaystyle\left(F_{w^{-1/m^{\prime}}(a_{j})}^{m_{j}}(v(i)^{1/m^{\prime}})\right)^{m}$
$\displaystyle=\sum_{j=k+1}^{q}\sum_{l=0}^{N}\big{[}v_{l}(i)\big{]}^{m/m^{\prime}}\alpha_{j,l}\frac{w_{n_{j}-n_{k}+l+1}(a_{k})\cdots
w_{n_{j}+l}(a_{k})}{w_{l+1}(a_{j})\cdots
w_{n_{j}+l}(a_{j})}e_{n_{j}-n_{k}+l}.$

Hence, (II) follows from the unconditionality of $(e_{n})$ if $\varepsilon$ is
small enough

In order to prove (III) and (IV), we take $\lambda=(a,b)\in\Lambda_{k}$, for
some $k=0,...,q$, we fix $m\in[m^{\prime},m^{\prime\prime}]$ and we write, for
$i=1,2$,

$\displaystyle
B_{w(a)}^{m_{k}}\left(F_{w^{-1/m^{\prime}}(a_{k})}^{m_{k}}(v(i))^{1/m^{\prime}}\right)^{m}$
$\displaystyle=\sum_{l=0}^{N}v_{l}(i)^{m/m^{\prime}}\frac{w_{l+1}(a)\cdots
w_{n_{k}-(N-l)}(a)}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{m/m^{\prime}}}e_{l}.$

Condition (III) corresponds to the case $m>m^{\prime}$, for which we write

$\displaystyle\frac{w_{l+1}(a)\cdots
w_{n_{k}-(N-l)}(a)}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{m/m^{\prime}}}$
$\displaystyle\leq\frac{1}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{m/m^{\prime}-1}}$
$\displaystyle=z_{k,l}\frac{1}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{1/m^{\prime}}},$

where

$z_{k,l}=\frac{1}{\big{[}w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})\big{]}^{\frac{m-1}{m^{\prime}}-1}}.$

Once again, since $X$ has a continuous norm, from (ii) and as
$\frac{m-1}{m^{\prime}}-1\geq 0$, we get that $(z_{k,l})$ is bounded uniformly
on $k$ and $l$. Now we can use the fact that $(e_{n})$ is bounded bellow and
repeat what we did with (3), now with $(x_{k,l})$ having only one non-zero
term. Hence (III) follows if we choose $\varepsilon>0$ small enough.

Finally, condition (IV) corresponds to the case $m=m^{\prime}$, in which we
write, for $i=1,2$,

$\displaystyle\left\|B_{w(a)}^{m_{k}}\left(F_{w^{-1/m^{\prime}}(a_{k})}^{m_{k}}(v(i))^{1/m^{\prime}}\right)^{m^{\prime}}-v(i)\right\|$
$\displaystyle\quad\quad\quad\leq\sum_{l=0}^{N}(|v_{l}(i)+1|)\left\|\left(\frac{w_{l+1}(a)\cdots
w_{n_{k}-(N-l)}(a)}{w_{l+1}(a_{k})\cdots
w_{n_{k}-(N-l)}(a_{k})}-1\right)e_{l}\right\|$

and it is easy to show that this becomes smaller than $\varepsilon$ provided
$\tau>0$ is taken very small and using $|f_{n}(a)-f_{n}(b)|\leq CF(n)|a-b|.$ ∎

If we restrict the $m$-th power in the statement to $m=1$, we obtain a
characterization of families admitting a common hypercyclic _vector_. Hence,
what distinguish vectors from algebras as far as this characterization goes is
the presence of the power $1/m$ in (ii). As soon as these conditions are
equivalent for a specific family of weights, one has that the family admits a
common hypercyclic vector if and only if it admits a common hypercyclic
algebra. We then get the following result.

######  Proposition 2.8.

Let $d\geq 1$, $\alpha\in(0,1]$, $w_{1}(a)\cdots w_{n}(a)=\exp(an^{\alpha})$
and $X=c_{0}(\mathbb{N})$ or $X=\ell_{p}(\mathbb{N}),p\in[1,+\infty)$. The
following assertions are equivalent:

  1. (a)

$(B_{w(\lambda(1))}\times\cdots\times B_{w(\lambda(d))})_{\lambda\in\Lambda}$
has a common hypercyclic vector in $X^{d}$;

  2. (b)

$(B_{w(\lambda(1))}\times\cdots\times B_{w(\lambda(d))})_{\lambda\in\Lambda}$
has a common hypercyclic algebra in $X^{d}$.

###### Proof.

The proof of $(b)\Rightarrow(a)$ is trivial. Let us assume $(a)$ and prove
$(b)$. This family then satisfy (c) of Theorem 2.7 for $m=1$. Let $m$ be
arbitrary, let $\tau>0$, let $N\geq 1$, $\varepsilon>0$ and let
$K\subset\Lambda$ compact. There is $N_{0}$ such that

(4) $\sum_{n=N_{0}}^{+\infty}\frac{1}{\exp(an^{\alpha})^{1/m}}<\varepsilon.$

From Theorem 2.7(c, $m=1$) there exist $(n_{k},\lambda_{k})_{k=1,...,q}$
satisfying $(i)$ and $(iii)$ of the same theorem. Condition $(ii)$ for this
arbitrary $m$ follows automatically from (4), what completes the proof. ∎

This particular case includes, for example, multiples of the backward shift.
In view of the results found in [4], we obtain the following.

######  Example 2.9.

Any family $\big{(}\lambda B\times\mu B\big{)}_{(\lambda,\mu)\in\Gamma}$
acting on $c_{0}(\mathbb{N})$ or $\ell_{p}(\mathbb{N}),p\in[1,+\infty),$ and
indexed by a Lipschitz curve $\Gamma\subset(1,+\infty)^{2}$ admits a common
hypercyclic vector. Hence, they admit as well common hypercyclic algebras for
the coordinatewise product.

##  3\. Convolution product

We will obtain Theorem 1.4 by applying the following more general key result
from [6], which is an adapted version of Birkoff’s hypercyclic theorem for
common hypercyclic algebras.

######  Proposition 3.1.

Let $\Lambda$ be a countable union of compact sets and let
$(T_{\lambda})_{\lambda\in\Lambda}$ be a family of operators in
$\mathcal{L}(X)$ such that the map $(\lambda,x)\mapsto T_{\lambda}(x)$ is
continuous from $\Lambda\times X$ into $X$. Assume that, for all compact sets
$K\subset\Lambda$, for all $m\geq 1$, for all $U,V$ non-empty open subsets of
$X$ and for all neighborhood $W$ of $0$, one can find $u\in U$ such that, for
all $\lambda\in K$, there exists $N\in\mathbb{N}$ satisfying

$\left\\{\begin{array}[]{l}\displaystyle T_{\lambda}^{N}(u^{n})\in W\textrm{
when }n\leq m-1,\\\ T_{\lambda}^{N}(u^{m})\in V.\end{array}\right.$

Then the set of points generating a common hypercyclic algebra for the family
$(T_{\lambda})_{\lambda\in\Lambda}$ is residual in $X$.

As for the covering, we didn’t manage to apply Lemma 2.6 to
$w_{n}(\lambda)=1+\frac{\lambda}{n}$. As this result was made having in mind
the case $w_{1}(a)\cdots w_{n}(a)=\exp(an^{\alpha})$, it well fits this
setting and naturally grants a common hypercyclic vector. The case
$w_{n}(\lambda)=1+\frac{\lambda}{n}$ is rather different. Now, the function
$a\mapsto\sum_{k=1}^{n}\log\big{(}w_{k}(a)\big{)}$ is $\log(n)$-Lipschitz and
a partition made of cubes of side $\frac{\tau}{n_{k}^{\alpha}}$ doesn’t seem
to work well. To overcome this apparent incompatibility we will insist on
defining cubes of side $\frac{\tau}{\log(n_{k})}$ instead.

In one dimension, we could define something like
$\lambda_{j+1}=\lambda_{j}+\frac{\tau}{\log(n_{k})}$ and proceed with similar
calculations as in [6, Theorem 3.12]. The tricky part being to define a
suitable sequence $(n_{k})_{k}$, but once it is done, this gives a common
hypercyclic algebra for $\big{(}B_{w(\lambda)}\big{)}_{\lambda>0}$ and answer
[6, Question 3.16] in the affirmative. What interest us, however, is the
possibility of finding such an algebra for
$\big{(}B_{w(\lambda_{1})}\times\cdots\times
B_{w(\lambda_{d})}\big{)}_{\lambda\in(0,+\infty)^{d}}$ for any $d\geq 1$.

As we have done before, we will prove the result for $d=2$, but it is clear
that the same can be done for any $d\geq 1$.

###### Proof of Theorem 1.4.

Let $\big{(}T_{\lambda,\mu}\big{)}_{(\lambda,\mu)\in(0,+\infty)^{2}}$ defined
by $T_{\lambda,\mu}:=B_{w(\lambda)}\oplus B_{w(\mu)}$ on $X\times X$, where
$X=\ell_{1}(\mathbb{N})$ and $w_{n}(\lambda)=1+\frac{\lambda}{n}$. We aim to
apply Proposition 3.1 to this family.

We fix
$[a^{\prime},b^{\prime}]\times[a^{\prime\prime},b^{\prime\prime}]\subset(0,+\infty)^{2}$
with $b^{\prime}<2a^{\prime}$ and $b^{\prime\prime}<2a^{\prime\prime}$ and let
$m\in\mathbb{N}$. We also fix an integer
$r>\max\\{\frac{1}{a^{\prime}},\frac{1}{a^{\prime\prime}},1\\}$ (as we will
see, one could just take $r=1$ in the case
$\min\\{a^{\prime},a^{\prime\prime}\\}>1$). As the products $w_{1}(x)\cdots
w_{n}(x)$ behave like $n^{x}$, is it enough to prove the claim for
$w_{1}(x)\cdots w_{n}(x)=n^{x}$. This will simplify the proof as there will be
less constant factors to be dealt with.

For $m=1$ the proposition is equivalent to the existence of a common
hypercyclic vector (which is known to be true from [4]). Let us assume $m\geq
2$. Let $U,V,W\subset X\times X$ open and non-empty, with $0\in W$. Let
$(u_{1},u_{2})\in U$, $(v_{1},v_{2})\in V$, $p\in\mathbb{N}$ and $\eta>0$ such
that $B\big{(}(u_{1},u_{2});2\eta\big{)}\subset U$,
$B\big{(}(v_{1},v_{2});3\eta\big{)}\subset V$ and
$\max\text{supp}(u_{1},u_{2},v_{1},v_{2})\leq p$. Since for each
$n\in\mathbb{N}$ the function $a\mapsto\sum_{k=1}^{n}\log(w_{k}(a))$ is
$\log(n)$-Lipschitz and $p$ is fixed, there exists a constant $C_{v}$ such
that, for all $l=0,...,p$ and all $a,b>0$,

(5)
$\Bigg{|}\sum_{j=l+1}^{n+l}\log\big{(}w_{j}(a)\big{)}-\sum_{j=l+1}^{n+l}\log\big{(}w_{j}(b)\big{)}\Bigg{|}\leq
C_{v}\log(n)|a-b|.$

Let us write $v_{i}=\sum_{l=0}^{p}v_{i,l}e_{l}$ for $i=1,2$ and choose
$\sigma$ a very big integer of the form $n^{m}$ (conditions on its size
later). We can choose $\sigma$ such that $\lfloor(\log\sigma)^{3}+1\rfloor$ is
the square of some integer $q$. We then have $\sqrt{q}\in\mathbb{N}$ and

$(\log\sigma)^{3}\leq q\leq(\log\sigma)^{3}+1.$

We now define $a=\min\\{a^{\prime},a^{\prime\prime}\\}$,
$b=\max\\{b^{\prime},b^{\prime\prime}\\}$, and we consider a covering of
$[a,b]^{2}$ by $q$ squares $\Lambda_{1},...,\Lambda_{q}$ of side
$\frac{b-a}{\sqrt{q}}$ and centers
$(\lambda_{1},\mu_{1}),...,(\lambda_{q},\mu_{q})$. We choose the sequence of
powers

$N_{j}=(m-1)\sigma+\sigma^{\frac{m-1}{m}}(j+1)^{r},\quad j=1,...,q.$

Since $\sigma$ is of the form $n^{m}$, the power $N_{j}$ is a positive integer
for all $j=1,...,q$. Let

$(u^{\prime},u^{\prime\prime})=(u_{1},u_{2})+\sum_{j=1}^{q}\sum_{l=0}^{p}\big{(}d_{l,j}^{\prime}e_{N_{j}-(m-1)\sigma+l},d_{l,j}^{\prime\prime}e_{N_{j}-(m-1)\sigma+l}\big{)}+(\varepsilon_{1}e_{\sigma},\varepsilon_{2}e_{\sigma}),$

where

$d_{l,j}^{\prime}:=\frac{v_{1,l}}{m\varepsilon_{1}^{m-1}w_{l+1}(\lambda_{j})\cdots
w_{l+N_{j}}(\lambda_{j})},\quad
d_{l,j}^{\prime\prime}:=\frac{v_{2,l}}{m\varepsilon_{2}^{m-1}w_{l+1}(\mu_{j})\cdots
w_{l+N_{j}}(\mu_{j})},$
$\varepsilon_{1}:=\left(\frac{1}{w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})}\right)^{\frac{1}{m}},\quad\varepsilon_{2}:=\left(\frac{1}{w_{1}(a^{\prime\prime})\cdots
w_{m\sigma}(a^{\prime\prime})}\right)^{\frac{1}{m}}.$

Let us first show that $(u^{\prime},u^{\prime\prime})\in U$ if $\sigma$ is big
enough. We can calculate it coordinate by coordinate and get, for some
constant $C_{v}^{\prime}>0$ depending on $v$,

$\displaystyle\|u^{\prime}-u_{1}\|_{1}$
$\displaystyle\leq\left\|\sum_{j=1}^{q}\sum_{l=0}^{p}d_{l,j}^{\prime}e_{N_{j}-(m-1)\sigma+l}\right\|_{1}+\|\varepsilon_{1}e_{\sigma}\|_{1}$
$\displaystyle\leq\sum_{j=1}^{q}\sum_{l=0}^{p}\frac{|v_{1,l}|}{m}\times\frac{\big{[}w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})\big{]}^{\frac{m-1}{m}}}{w_{l+1}(\lambda_{j})\cdots
w_{l+N_{j}}(\lambda_{j})}+\varepsilon_{1}$ $\displaystyle\leq
C_{v}^{\prime}\sum_{j=1}^{q}\frac{\big{(}w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})\big{)}^{\frac{m-1}{m}}}{w_{1}(a^{\prime})\cdots
w_{N_{j}}(a^{\prime})}+\Bigg{(}\frac{1}{w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})}\Bigg{)}^{\frac{1}{m}}$
$\displaystyle=C_{v}^{\prime}\sum_{j=1}^{q}\left(\frac{(m\sigma)^{\frac{m-1}{m}}}{(m-1)\sigma+\sigma^{\frac{m-1}{m}}(j+1)^{r}}\right)^{a^{\prime}}+\Bigg{(}\frac{1}{w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})}\Bigg{)}^{\frac{1}{m}}$ $\displaystyle\leq
C_{v}^{\prime}\sum_{j=1}^{+\infty}\left(\frac{m^{\frac{m-1}{m}}}{(m-1)\sigma^{1/m}+(j+1)^{r}}\right)^{a^{\prime}}+\Bigg{(}\frac{1}{w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})}\Bigg{)}^{\frac{1}{m}},$

and this goes to zero as $\sigma\to+\infty$, hence $\|u^{\prime}-u_{1}\|<\eta$
if $\sigma$ is big enough. Analogously $\|u^{\prime\prime}-u_{2}\|<\eta$, what
proves that $(u^{\prime},u^{\prime\prime})\in U$.

Let us now show that, for all
$(\lambda,\mu)\in[a^{\prime},b^{\prime}]\times[a^{\prime\prime},b^{\prime\prime}]$,
there is $N$ satisfying

(6) $\left\\{\begin{array}[]{l}\displaystyle
T_{\lambda,\mu}^{N}\big{(}(u^{\prime},u^{\prime\prime})^{n}\big{)}\in
W\textrm{ for }n<m,\\\
T_{\lambda,\mu}^{N}\big{(}(u^{\prime},u^{\prime\prime})^{m}\big{)}\in
V.\end{array}\right.$

Given
$(\lambda,\mu)\in[a^{\prime},b^{\prime}]\times[a^{\prime\prime},b^{\prime\prime}]$,
let $i\in\\{1,...,q\\}$ such that $(\lambda,\mu)\in\Lambda_{i}.$ We then
choose $N=N_{i}$. Let us verify the conditions in (6).

We first notice that, if $n<m$, since
$\max\text{supp}(u^{\prime},u^{\prime\prime})^{n}\leq(m-1)\sigma<N_{i}=N$ if
$\sigma$ is big enough, we have
$T_{\lambda,\mu}^{N}\big{(}(u^{\prime},u^{\prime\prime})^{n}\big{)}=0\in W.$
For the $m$-th power we get

$\displaystyle(u^{\prime},u^{\prime\prime})^{m}$
$\displaystyle=(P_{0}^{\prime},P_{0}^{\prime\prime})+\sum_{j=1}^{q}\sum_{l=0}^{p}\big{(}m\varepsilon_{1}^{m-1}d_{l,j}^{\prime}e_{N_{j}+l},m\varepsilon_{2}^{m-1}d_{l,j}^{\prime\prime}e_{N_{j}+l}\big{)}+(\varepsilon_{1}^{m}e_{m\sigma},\varepsilon_{2}^{m}e_{m\sigma}),$

where

$\text{supp}(P_{0}^{\prime},P_{0}^{\prime\prime})\subset[0,(m-1)\sigma+p]\cup[0,(m-2)\sigma+2(N_{q}-(m-1)\sigma+p)].$

We aim to apply $T_{\lambda,\mu}^{N_{i}}$ on
$(u^{\prime},u^{\prime\prime})^{m}$, many parcels will be eliminated by the
support. If $\sigma$ is big enough, we’ll have $(m-1)\sigma+p<N_{i}$.
Moreover,

$\displaystyle N_{i}-\big{[}(m-2)\sigma+2(N_{q}-(m-1)\sigma+p)\big{]}\geq
N_{0}-\big{[}(m-2)\sigma+2(N_{q}-(m-1)\sigma+p)\big{]}=$
$\displaystyle=(m-1)\sigma+\sigma^{\frac{m-1}{m}}-(m-2)\sigma-2\big{(}(m-1)\sigma+\sigma^{\frac{m-1}{m}}(q+1)^{r}-(m-1)\sigma+p\big{)}=$
$\displaystyle\geq\sigma+\sigma^{\frac{m-1}{m}}-2\Big{(}\sigma^{\frac{m-1}{m}}\Big{(}\big{[}\log(\sigma)\big{]}^{3}+2\Big{)}^{r}+p\Big{)}\geq\sigma-2C_{r}\sigma^{\frac{m-1}{m}}\big{[}\log(\sigma)\big{]}^{3r},$

for some constant $C_{r}>0$ depending on $r$. Then we will have
$N_{i}>\max\text{supp}(P_{0})$ if $\sigma$ is taken big enough. Furthermore,
in the sum
$\sum_{j=1}^{q}\sum_{l=0}^{p}\big{(}m\varepsilon_{1}^{m-1}d_{l,j}^{\prime}e_{N_{j}+l},m\varepsilon_{2}^{m-1}d_{l,j}^{\prime\prime}e_{N_{j}+l}\big{)}$,
we can use that $N_{i}>N_{j}+p$ when $j<i$ to conclude that all parcels
indexed by $j=1,...,i-1$ have maximum support less than $N_{i}$. The parcel
$j=i$ is the one we will use to approach $(v_{1},v_{2})$ and the final parcels
$j=i+1,...,q$, as well as the separating term
$(\varepsilon^{m}_{1}e_{m\sigma},\varepsilon^{m}_{2}e_{m\sigma})$, will be
handled in our next calculations. All in all, we write

$\displaystyle T_{\lambda,\mu}^{N_{i}}\big{(}(u^{\prime},u^{\prime\prime}$
$\displaystyle)^{m}\big{)}=(P_{1}^{\prime},P_{1}^{\prime\prime})+(P_{2}^{\prime},P_{2}^{\prime\prime})+(P_{3}^{\prime},P_{3}^{\prime\prime}),$

where

$\displaystyle(P_{1}^{\prime},P_{1}^{\prime\prime})$
$\displaystyle=\sum_{l=0}^{p}\Big{(}\frac{w_{l+1}(\lambda)\cdots
w_{l+N_{i}}(\lambda)}{w_{l+1}(\lambda_{i})\cdots
w_{l+N_{i}}(\lambda_{i})}v_{1,l}e_{l},\frac{w_{l+1}(\mu)\cdots
w_{l+N_{i}}(\mu)}{w_{l+1}(\mu_{i})\cdots
w_{l+N_{i}}(\mu_{i})}v_{2,l}e_{l}\Big{)},$ $\displaystyle P_{2}^{\prime}$
$\displaystyle=\sum_{j=i+1}^{q}\sum_{l=0}^{p}\frac{w_{N_{j}-N_{i}+l+1}(\lambda)\cdots
w_{l+N_{j}}(\lambda)}{w_{l+1}(\lambda_{j})\cdots
w_{l+N_{j}}(\lambda_{j})}v_{1,l}e_{N_{j}-N_{i}+l},$ $\displaystyle
P_{2}^{\prime\prime}$
$\displaystyle=\sum_{j=i+1}^{q}\sum_{l=0}^{p}\frac{w_{N_{j}-N_{i}+l+1}(\mu)\cdots
w_{l+N_{j}}(\mu)}{w_{l+1}(\mu_{j})\cdots
w_{l+N_{j}}(\mu_{j})}v_{2,l}e_{N_{j}-N_{i}+l},$ $\displaystyle P_{3}^{\prime}$
$\displaystyle=\varepsilon_{1}^{m}w_{m\sigma-N_{i}+1}(\lambda)\cdots
w_{m\sigma}(\lambda)e_{m\sigma-N_{i}},$ $\displaystyle P_{3}^{\prime\prime}$
$\displaystyle=\varepsilon_{2}^{m}w_{m\sigma-N_{i}+1}(\lambda)\cdots
w_{m\sigma}(\lambda)e_{m\sigma-N_{i}}.$

The proof will be finished if we manage to prove the following:

  1. (I)

$\|(P_{1}^{\prime},P_{1}^{\prime\prime})-(v_{1},v_{2})\|_{1}<\eta$;

  2. (II)

$\|(P_{2}^{\prime},P_{2}^{\prime\prime})\|_{1}<\eta$;

  3. (III)

$\|(P_{3}^{\prime},P_{3}^{\prime\prime})\|_{1}<\eta$.

Let us begin with (I) and show that

$\|(P_{1}^{\prime},P_{1}^{\prime\prime})-(v_{1},v_{2})\|_{1}\leq\|P_{1}^{\prime}-v_{1}\|_{1}+\|P_{1}^{\prime\prime}-v_{2}\|_{1}<\eta.$

First we notice that, from (5), for all $l=0,...,p$,

$\displaystyle\Big{|}\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda)\big{)}-\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda_{i})\big{)}\Big{|}$
$\displaystyle\leq C_{v}\log(N_{i})|\lambda-\lambda_{i}|$ $\displaystyle\leq
C_{v}(b-a)\frac{\log(N_{i})}{\sqrt{q}}$ $\displaystyle\leq
C_{v}(b-a)\frac{\log(N_{i})}{\sqrt{\log(\sigma)^{3}}}\xrightarrow[]{\sigma\to+\infty}0.$

Then, if $\sigma$ is big enough, we get

(7)
$\Big{|}\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda)\big{)}-\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda_{i})\big{)}\Big{|}<\min\left(1,\frac{\eta}{4(p+1)\|(v_{1},v_{2})\|_{\infty}}\right).$

In particular, the absolute value in the left-hand side is smaller than 1, so
we can use the estimate $|\exp(x)-1|\leq 2|x|$ which holds for all
$x\in[-1,1]$. By using (7) we then obtain

$\displaystyle\|P_{1}^{\prime}-v_{1}\|_{1}$
$\displaystyle=\left\|\sum_{l=0}^{p}\Big{(}\frac{w_{l+1}(\lambda)\cdots
w_{l+N_{i}}(\lambda)}{w_{l+1}(\lambda_{i})\cdots
w_{l+N_{i}}(\lambda_{i})}-1\Big{)}v_{1,l}e_{l}\right\|_{1}$
$\displaystyle\leq\sum_{l=0}^{p}|v_{1,l}|\Bigg{|}\exp\Big{(}\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda)\big{)}-\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda_{i})\big{)}\Big{)}-1\Bigg{|}$
$\displaystyle\leq\sum_{l=0}^{p}|v_{1,l}|\cdot
2\Big{|}\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda)\big{)}-\sum_{j=l+1}^{N_{i}+l}\log\big{(}w_{j}(\lambda_{i})\big{)}\Big{|}$
$\displaystyle<\sum_{l=0}^{p}|v_{1,l}|\cdot
2\frac{\eta}{4(p+1)\|v_{1}\|_{\infty}}$ $\displaystyle<\frac{\eta}{2}.$

Analogously we find $\|P_{1}^{\prime\prime}-v_{2}\|_{1}<\frac{\eta}{2}$, what
verifies (I).

For (II), we first notice that, for some constant $C_{v}^{\prime\prime}>0$
depending on $v$,

$\displaystyle\Big{\|}\sum_{l=0}^{p}\frac{w_{N_{j}-N_{i}+l+1}(\lambda)\cdots
w_{l+N_{j}}(\lambda)}{w_{l+1}(\lambda_{j})\cdots
w_{l+N_{j}}(\lambda_{j})}v_{1,l}e_{N_{j}-N_{i}+l}\Big{\|}_{1}$
$\displaystyle\leq\sum_{l=0}^{p}|v_{1,l}|\frac{w_{N_{j}-N_{i}+l+1}(\lambda)\cdots
w_{N_{j}+l}(\lambda)}{w_{l+1}(\lambda_{j})\cdots w_{N_{j}+l}(\lambda_{j})}$
$\displaystyle\leq
C_{v}^{\prime\prime}\frac{(N_{j}+l)^{\lambda-\lambda_{j}}}{(N_{j}-N_{i}+l)^{\lambda}}.$

We plan to show that this is the general term of a convergent series on $j$
whose sum can be made small by increasing $\sigma$. As we are dealing with a
bi-dimensional partition, it is necessary to keep track of the sign of
$|\lambda-\lambda_{j}|$. If $\lambda_{j}\geq\lambda$ then, just as in one
dimension, we have $(N_{j}+l)^{\lambda-\lambda_{j}}\leq 1$, that is, our
problem reduces to estimate

$\frac{1}{(N_{j}-N_{i}+l)^{\lambda}},$

what is included in the calculations below. Let us now consider the more
difficult case $\lambda_{j}<\lambda$, which only occurs in two (or more)
dimensions. We have

$\displaystyle\frac{(N_{j}+l)^{\lambda-\lambda_{j}}}{(N_{j}-N_{i}+l)^{\lambda}}$
$\displaystyle=\frac{\big{(}(m-1)\sigma+\sigma^{\frac{m-1}{m}}(j+1)^{r}+l\big{)}^{\lambda-\lambda_{j}}}{\Big{(}\big{(}(j+1)^{r}-(i+1)^{r}\big{)}\sigma^{\frac{m-1}{m}}+l\Big{)}^{\lambda}}$
$\displaystyle\overset{l\geq
0}{\leq}\frac{\sigma^{\lambda-\lambda_{j}}(j+1)^{r(\lambda-\lambda_{j})}\big{(}\frac{(m-1)}{(j+1)^{r}}+\sigma^{\frac{-1}{m}}+\frac{l}{\sigma(j+1)^{r}}\big{)}^{\lambda-\lambda_{j}}}{\sigma^{\lambda\frac{m-1}{m}}(j+1)^{r\lambda}\big{(}1-\frac{(i+1)^{r}}{(j+1)^{r}}\big{)}^{\lambda}}$
$\displaystyle\overset{r\geq
1}{\leq}\frac{\big{(}\frac{(m-1)}{(j+1)^{r}}+\sigma^{\frac{-1}{m}}+\frac{l}{\sigma(j+1)^{r}}\big{)}^{\lambda-\lambda_{j}}}{\sigma^{\lambda_{j}-\frac{\lambda}{m}}(j+1)^{r\lambda_{j}}\big{(}1-\frac{i+1}{j+1}\big{)}^{\lambda}}$
$\displaystyle\\!\\!\overset{j\geq
i+1}{\leq}\frac{C_{0}}{\sigma^{\lambda_{j}-\frac{\lambda}{m}}(j+1)^{r\lambda_{j}}\big{(}1-\frac{i+1}{i+2}\big{)}^{\lambda}}$
$\displaystyle\overset{i\leq
q}{\leq}\frac{C_{0}}{(j+1)^{r\lambda_{j}}\Big{(}\frac{\sigma^{\frac{\lambda_{j}-\frac{\lambda}{m}}{\lambda}}}{q+2}\Big{)}^{\lambda}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\overset{q\leq[\log(\sigma)]^{3}+1}{\leq}{\frac{C_{0}}{(j+1)^{r\lambda_{j}}\Big{(}\frac{\sigma^{c^{\prime}}}{(\log\sigma)^{3}+3}\Big{)}^{\lambda}}},$

for some constant $C_{0}$ depending on $a,b,r,m,p$ and
$c^{\prime}=\frac{a^{\prime}}{b^{\prime}}-\frac{1}{m}\leq\frac{\lambda_{j}}{\lambda}-\frac{1}{m}$
is positive since $m\geq 2$ and $2a^{\prime}\geq b^{\prime}$. These estimates
show that

$\displaystyle\|P_{2}^{\prime}\|_{1}$
$\displaystyle=\left\|\sum_{j=i+1}^{q}\sum_{l=0}^{p}\frac{w_{N_{j}-N_{i}+l+1}(\lambda)\cdots
w_{l+N_{j}}(\lambda)}{w_{l+1}(\lambda_{j})\cdots
w_{l+N_{j}}(\lambda_{j})}v_{1,l}e_{N_{j}-N_{i}+l}\right\|_{1}$
$\displaystyle\leq\frac{C_{v}^{\prime\prime}C_{0}}{\Big{(}\frac{\sigma^{c^{\prime}}}{(\log\sigma)^{3}+3}\Big{)}^{\lambda}}\sum_{j=i+1}^{q}\frac{1}{(j+1)^{ra^{\prime}}}\xrightarrow[]{\sigma\to+\infty}0.$

Hence, $\|P_{2}^{\prime}\|_{1}<\frac{\eta}{2}$ if $\sigma$ is big enough.
Analogously we get $\|P_{2}^{\prime\prime}\|_{1}<\frac{\eta}{2}$, what shows
(II).

Let us proceed to the proof of (III). We have

$\displaystyle\|P_{3}^{\prime}\|_{1}$
$\displaystyle=\|\varepsilon_{1}^{m}w_{m\sigma-N_{i}+1}(\lambda)\cdots
w_{m\sigma}(\lambda)e_{m\sigma-N_{i}}\|_{1}$
$\displaystyle\leq\frac{w_{m\sigma-N_{q}+1}(b^{\prime})\cdots
w_{m\sigma}(b^{\prime})}{w_{1}(a^{\prime})\cdots w_{m\sigma}(a^{\prime})}$
$\displaystyle=\frac{w_{1}(b^{\prime})\cdots
w_{m\sigma}(b^{\prime})}{w_{1}(a^{\prime})\cdots
w_{m\sigma}(a^{\prime})}\times\frac{1}{w_{1}(b^{\prime})\cdots w_{m\sigma-
N_{q}}(b^{\prime})}$
$\displaystyle\leq\frac{(m\sigma)^{b^{\prime}-a^{\prime}}}{(m\sigma-
N_{q})^{b^{\prime}}}$
$\displaystyle\leq{\frac{(m\sigma)^{b^{\prime}-a^{\prime}}}{(\sigma-\sigma^{\frac{m-1}{m}}((\log\sigma)^{3}+2)^{r})^{b^{\prime}}}}\xrightarrow[]{\sigma\to+\infty}0.$

Hence, $\|P_{3}^{\prime}\|_{1}<\frac{\eta}{2}$ if $\sigma$ is big enough.
Analogously, $\|P_{3}^{\prime\prime}\|_{1}<\frac{\eta}{2}$ and the proof is
done. ∎

We finish this paper by returning to the one dimensional framework. Let $I$ be
an interval and let $\lambda\in I\mapsto\big{(}w_{n}(\lambda)\big{)}_{n}$ be a
continuously parametrized family of weights such that
$\lambda\mapsto\sum_{k=1}^{n}\log\big{(}w_{k}(\lambda)\big{)}$ is
$F(n)$-Lipschitz for some function $F:\mathbb{N}\to\mathbb{N}$. As exemplified
by its authors, [6, Theorem 3.12] can be applied for some cases where
$F(n)=n$. It turns out that it can be more generally applied for the case
where $w_{1}(\lambda)\cdots w_{n}(\lambda)=\exp(\lambda n^{\alpha})$ with
$\alpha\in(0,1]$, which is a particular case where $F(n)=n^{\alpha}$. However,
the same result doesn’t work for the case $F(n)=\log(n)$, although
$w_{n}(\lambda)=1+\frac{\lambda}{n}$ does admit a common hypercyclic algebra
as we proved here (through similar, yet different, calculations).

######  Question 3.2.

Could one obtain a unified one dimensional criterion on $F(n)$ granting a
common hypercyclic subalgebra of $\ell_{1}(\mathbb{N})$ for the convolution
product and including $F(n)=n^{\alpha},\alpha\leq 1,$ and $F(n)=\log(n)$ as
particular cases?

Acknowledgement. The author would like to thank the referee for their
thoughtful comments and efforts towards improving the manuscript.

## References

  * [1] E. Abakumov and J. Gordon.  Common hypercyclic vectors for multiples of backward shift.  J. Funct. Anal. 200 (2003), 494–504.
  * [2] R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda. Powers of hypercyclic functions for some classical hypercyclic operators.  Integral Equations Operator Theory 58(4) (2007), 591–596.
  * [3] R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda. Sums and products of bad functions. In Function spaces, volume 435 of _Contemp. Math._ , pages 47–52. Amer. Math. Soc., Providence, RI, 2007.
  * [4] F. Bayart, F. Costa Jr. and Q. Menet. Common hypercyclic vectors and dimension of the parameter set.  Available at https://arxiv.org/abs/2103.13152.
  * [5] F. Bayart, F. Costa Júnior and D. Papathanasiou.  Baire theorem and hypercyclic algebras.  Advances in Mathematics 376 (2021), 107419.
  * [6] F. Bayart, F. Costa Júnior and D. Papathanasiou.  Common and disjoint hypercyclic algebras.  To appear at _Israel Journal of Mathematics_.
  * [7] F. Bayart and É. Matheron.  Dynamics of linear operators, volume 179 of Cambridge Tracts in Math.  Cambridge University Press, 2009.
  * [8] G. Costakis and M. Sambarino.  Genericity of wild holomorphic functions and common hypercyclic vectors.  Adv. Math. 182 (2004), 278–306.
  * [9] K.-G. Grosse-Erdmann and A. Peris.  Linear chaos.  Springer, 2011.
  * [10] S. Shkarin. On the set of hypercyclic vectors for the differentiation operator.  Israel J. Math. 180 (2010), 271–283.


To subleading order, we have

$\displaystyle
H^{+(2)}_{\mu\nu}(\omega,\Omega)=H^{-(2)}_{\mu\nu}(\omega,\Omega^{P}).$ (D.28)

#### Electric and magnetic fields

For linearized gravity, we could define electric and magnetic fields analogous
to Maxwell theory

$\displaystyle{\mathcal{E}}_{mn}=-R_{0m0n},\qquad{\mathcal{B}}_{mn}=\frac{1}{2}\,\epsilon_{npq}\,R_{0m}^{\;\;\;\;pq},\qquad
m,n=1,2,3$ (D.29)

With (3.102), we can write them explicitly

$\displaystyle{\mathcal{E}}_{mn}=\frac{1}{2}(\partial_{0}^{2}h_{mn}-\partial_{0}\partial_{m}h_{0n}-\partial_{0}\partial_{n}h_{0m}+\partial_{n}\partial_{m}h_{00}),$
(D.30)
$\displaystyle{\mathcal{B}}_{mn}=-\frac{1}{2}\epsilon_{n}{}^{pq}(\partial_{p}\partial_{0}h_{mq}-\partial_{p}\partial_{m}h_{0q}).$
(D.31)

We may expand the electric and magnetic part asymptotically as

$\displaystyle\mathcal{E}_{mn}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{E}_{mn}^{+}(u,\Omega)}{r}+\sum_{k=2}^{\infty}\frac{\mathcal{E}_{mn}^{+(k)}(u,\Omega)}{r^{k}},$
(D.32) $\displaystyle\mathcal{B}_{mn}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{B}_{mn}^{+}(u,\Omega)}{r}+\sum_{k=2}^{\infty}\frac{\mathcal{B}_{mn}^{+(k)}(u,\Omega)}{r^{k}},$
(D.33) $\displaystyle\mathcal{E}_{mn}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{E}_{mn}^{-}(v,\Omega)}{r}+\sum_{k=2}^{\infty}\frac{\mathcal{E}_{mn}^{-(k)}(v,\Omega)}{r^{k}},$
(D.34) $\displaystyle\mathcal{B}_{mn}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{B}_{mn}^{-}(v,\Omega)}{r}+\sum_{k=2}^{\infty}\frac{\mathcal{B}_{mn}^{-(k)}(v,\Omega)}{r^{k}}.$
(D.35)

Sending to null infinity, we obtain

$\displaystyle{\mathcal{E}}^{+}_{mn}(u,\Omega)=\frac{1}{2}[\ddot{H}^{+}_{mn}(u,\Omega)+n_{m}(\Omega)\ddot{H}^{+}_{0n}(u,\Omega)+n_{n}(\Omega)\ddot{H}^{+}_{0m}(u,\Omega)+n_{m}(\Omega)n_{n}(\Omega)\ddot{H}^{+}_{00}(u,\Omega)],$
(D.36)
$\displaystyle{\mathcal{B}}^{+}_{mn}(u,\Omega)=\frac{1}{2}\epsilon_{n}{}^{pq}[n_{p}(\Omega)\ddot{H}^{+}_{mq}(u,\Omega)+n_{p}(\Omega)n_{m}(\Omega)\ddot{H}^{+}_{0q}(u,\Omega)]$
(D.37)

for $\mathcal{I}^{+}$, and

$\displaystyle{\mathcal{E}}^{-}_{mn}(v,\Omega)=\frac{1}{2}[\ddot{H}^{-}_{mn}(v,\Omega)-n_{m}(\Omega)\ddot{H}^{-}_{0n}(v,\Omega)-n_{n}(\Omega)\ddot{H}^{-}_{0m}(v,\Omega)+n_{m}(\Omega)n_{n}(\Omega)\ddot{H}^{-}_{00}(v,\Omega)],$
(D.38)
$\displaystyle{\mathcal{B}}^{-}_{mn}(v,\Omega)=-\frac{1}{2}\epsilon_{n}{}^{pq}[n_{p}(\Omega)\ddot{H}^{-}_{mq}(v,\Omega)-n_{p}(\Omega)n_{m}(\Omega)\ddot{H}^{-}_{0q}(v,\Omega)]$
(D.39)

for $\mathcal{I}^{-}$. Converting to Fourier space, one find

$\displaystyle{\mathcal{E}}^{+}_{mn}(\omega,\Omega)=-\frac{1}{2}\omega^{2}[H^{+}_{mn}(\omega,\Omega)+n_{m}(\Omega)H^{+}_{0n}(\omega,\Omega)+n_{n}(\Omega)H^{+}_{0m}(\omega,\Omega)+n_{m}(\Omega)n_{n}(\Omega)H^{+}_{00}(\omega,\Omega)],$
(D.40)
$\displaystyle{\mathcal{B}}^{+}_{mn}(\omega,\Omega)=-\frac{1}{2}\omega^{2}\epsilon_{n}{}^{pq}[n_{p}(\Omega)H^{+}_{mq}(\omega,\Omega)+n_{p}(\Omega)n_{m}(\Omega)H^{+}_{0q}(\omega,\Omega)],$
(D.41)

and

$\displaystyle{\mathcal{E}}^{-}_{mn}(\omega,\Omega)=-\frac{1}{2}\omega^{2}[H^{-}_{mn}(\omega,\Omega)-n_{m}(\Omega)H^{-}_{0n}(\omega,\Omega)-n_{n}(\Omega)H_{0m}(\omega,\Omega)+n_{m}(\Omega)n_{n}(\Omega)H^{-}_{00}(\omega,\Omega)],$
(D.42)
$\displaystyle{\mathcal{B}}^{-}_{mn}(\omega,\Omega)=\frac{1}{2}\omega^{2}\epsilon_{n}{}^{pq}[n_{p}(\Omega)H^{-}_{mq}(\omega,\Omega)-n_{p}(\Omega)n_{m}(\Omega)H^{-}_{0q}(\omega,\Omega)].$
(D.43)

Using the relation

$\displaystyle n_{i}(\Omega^{P})=-n_{i}(\Omega),$ (D.44)

we get the antipodal condition for electric and magnetic fields

$\displaystyle{\mathcal{E}}^{+}_{mn}(\omega,\Omega)=-{\mathcal{E}}^{-}_{mn}(\omega,\Omega^{P}),\qquad{\mathcal{B}}^{+}_{mn}(\omega,\Omega)=-{\mathcal{B}}^{-}_{mn}(\omega,\Omega^{P}).$
(D.45)

As a matter of fact, we could discuss the antipodal matching conditions for
linearized Riemann tensor which is more general than electric and magnetic
fields. We expand the Riemann tensor asymptotically as

$\displaystyle R_{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{R^{+}_{\mu\nu\rho\sigma}(u,\Omega)}{r}+\sum_{k=2}^{\infty}\frac{R^{+(k)}_{\mu\nu\rho\sigma}(u,\Omega)}{r^{k}},$
(D.46) $\displaystyle R_{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{R^{-}_{\mu\nu\rho\sigma}(v,\Omega)}{r}+\sum_{k=2}^{\infty}\frac{R^{+(k)}_{\mu\nu\rho\sigma}(v,\Omega)}{r^{k}}.$
(D.47)

At future/past null infinity, we find

$\displaystyle R^{+}_{\mu\nu\rho\sigma}(u,\Omega)=$
$\displaystyle\frac{1}{2}\left(n_{\rho}n_{\nu}\ddot{H}^{+}_{\rho\sigma}(u,\Omega)-n_{\rho}n_{\mu}\ddot{H}^{+}_{\nu\sigma}(u,\Omega)-n_{\sigma}n_{\nu}\ddot{H}^{+}_{\mu\rho}(u,\Omega)+n_{\rho}n_{\sigma}\ddot{H}^{+}_{\nu\rho}(u,\Omega)\right),$
(D.48) $\displaystyle R^{-}_{\mu\nu\rho\sigma}(v,\Omega)=$
$\displaystyle\frac{1}{2}\left(n_{\rho}n_{\nu}\ddot{H}^{-}_{\rho\sigma}(v,\Omega)-n_{\rho}n_{\mu}\ddot{H}^{-}_{\nu\sigma}(v,\Omega)-n_{\sigma}n_{\nu}\ddot{H}^{-}_{\mu\rho}(v,\Omega)+n_{\rho}n_{\sigma}\ddot{H}^{-}_{\nu\rho}(v,\Omega)\right).$
(D.49)

In Fourier space, they are equivalent to

$\displaystyle R^{+}_{\mu\nu\rho\sigma}(\omega,\Omega)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\omega^{2}\left(n_{\rho}n_{\nu}H^{+}_{\rho\sigma}(\omega,\Omega)-n_{\rho}n_{\mu}H^{+}_{\nu\sigma}(\omega,\Omega)-n_{\sigma}n_{\nu}H^{+}_{\mu\rho}(\omega,\Omega)+n_{\rho}n_{\sigma}H^{+}_{\nu\rho}(\omega,\Omega)\right),$
$\displaystyle R^{-}_{\mu\nu\rho\sigma}(\omega,\Omega)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\omega^{2}\left(n_{\rho}n_{\nu}H^{-}_{\rho\sigma}(\omega,\Omega)-n_{\rho}n_{\mu}H^{-}_{\nu\sigma}(\omega,\Omega)-n_{\sigma}n_{\nu}H^{-}_{\mu\rho}(\omega,\Omega)+n_{\rho}n_{\sigma}H^{-}_{\nu\rho}(\omega,\Omega)\right).$

Consequently, we find the antipodal condition

$\displaystyle
R^{+}_{\mu\nu\rho\sigma}(\omega,\Omega)=-R^{-}_{\mu\nu\rho\sigma}(\omega,\Omega^{P}).$
(D.50)

###  D.2 Canonical quantization

In this subsection, we use mode expansion of quantized field to compute the
fundamental commutator of shear tensor in the transverse and traceless gauge.
The result will be same to the one in Bondi gauge.

Starting from (D.3), switching to retarded frame, and approaching future null
infinity, we find

$\displaystyle h_{AB}(t,\bm{x})=$
$\displaystyle\sum_{\alpha}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{\bm{k}}}}r^{2}Y^{i}_{A}Y^{j}_{B}[\epsilon^{*\alpha}_{ij}(\bm{k})b_{\alpha,\bm{k}}e^{-i\omega
t+i\bm{k}\cdot\bm{x}}+\epsilon^{\alpha}_{ij}(\bm{k})b^{\dagger}_{\alpha,\bm{k}}e^{i\omega
t-i\bm{k}\cdot\bm{x}}]$ $\displaystyle=$ $\displaystyle
r\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}\sum_{\ell
m}[c_{i,j;\omega,\ell,m}Y^{i}_{A}Y^{j}_{B}Y_{\ell,m}(\Omega)e^{-i\omega
u}+c^{\dagger}_{i,j;\omega,\ell,m}Y^{i}_{A}Y^{j}_{B}Y_{\ell,m}^{*}(\Omega)e^{i\omega
u}]+\mathcal{O}(1),$ (D.51)

where the boundary creation and annihilation operators are

$\displaystyle c_{i,j;\omega,\ell,m}=$
$\displaystyle\frac{\omega}{(2\pi)^{3/2}i}\int
d\Omega_{k}\sum_{\alpha}\epsilon_{ij}^{*\alpha}(\bm{k})b_{\alpha,\bm{k}}Y_{\ell,m}^{*}(\Omega_{k}),$
(D.52) $\displaystyle c^{\dagger}_{i,j;\omega,\ell,m}=$
$\displaystyle\frac{i\omega}{(2\pi)^{3/2}}\int
d\Omega_{k}\sum_{\alpha}\epsilon_{ij}^{\alpha}(\bm{k})b^{\dagger}_{\alpha,\bm{k}}Y_{\ell,m}(\Omega_{k}).$
(D.53)

One can insert back the coefficient $\sqrt{32\pi G}$ and read out the shear
tensor

$\displaystyle C_{AB}(u,\Omega)=\sqrt{32\pi
G}\int_{0}^{\infty}\frac{d\omega}{\sqrt{4\pi\omega}}\sum_{\ell
m}[c_{i,j;\omega,\ell,m}Y^{i}_{A}Y^{j}_{B}Y_{\ell,m}(\Omega)e^{-i\omega
u}+c^{\dagger}_{i,j;\omega,\ell,m}Y^{i}_{A}Y^{j}_{B}Y_{\ell,m}^{*}(\Omega)e^{i\omega
u}].$ (D.54)

From the completeness relation, we find

$\displaystyle[c_{i,j;\omega,\ell,m},c_{i^{\prime},j^{\prime};\omega^{\prime},\ell^{\prime},m^{\prime}}]=$
$\displaystyle[c^{\dagger}_{i,j;\omega,\ell,m},c^{\dagger}_{i^{\prime},j^{\prime};\omega^{\prime},\ell^{\prime},m^{\prime}}]=0,$
(D.55)
$\displaystyle[c_{i,j;\omega,\ell,m},c^{\dagger}_{i^{\prime},j^{\prime};\omega^{\prime},\ell^{\prime},m,}]=$
$\displaystyle\frac{1}{2}\delta(\omega-\omega^{\prime})\int
d\Omega\big{(}\bar{\eta}_{i,i^{\prime}}\bar{\eta}_{j,j^{\prime}}-\bar{\eta}_{i,j}\bar{\eta}_{i^{\prime},j^{\prime}}+\bar{\eta}_{i,j^{\prime}}\bar{\eta}_{i^{\prime},j}\big{)}Y_{\ell,m}^{*}(\Omega)Y_{\ell^{\prime},m^{\prime}}(\Omega).$
(D.56)

where

$\displaystyle\bar{\eta}_{ij}=\delta_{ij}-n_{i}n_{j}=Y_{i}^{A}(\Omega)Y_{jA}(\Omega).$
(D.57)

Thus we can obtain the following commutator

$\displaystyle[C_{AB}(u,\Omega),C_{CD}(u^{\prime},\Omega^{\prime})]=8\pi
GiP_{ACDB}\alpha(u-u^{\prime})\delta(\Omega-\Omega^{\prime}),$ (D.58)

as expected.

##  Appendix E Commutators

In this appendix, we will provide some details on the calculation of
commutators among flux operators. We will take
$[\mathcal{M}_{Y},\mathcal{M}_{Z}]$ as an example. To simplify computation, we
start from rewriting $\mathcal{M}_{Y}$ (without normal order written out due
to its irrelevance to non-central terms)

$\displaystyle\mathcal{M}_{Y}=$ $\displaystyle\frac{1}{32\pi G}\int dud\Omega
Y^{A}(u,\Omega)(\dot{C}^{BC}\nabla^{D}C^{EF}-C^{BC}\nabla^{D}\dot{C}^{EF})P_{ABCDEF}$
$\displaystyle=$ $\displaystyle\frac{1}{32\pi G}\int
dud\Omega\dot{C}^{BC}\Delta_{BC}(Y;C;u,\Omega),$ (E.1)

where $\Delta_{BC}(Y;C;u,\Omega)$ is given by (3.79). Then we can compute
straightforwardly

$\displaystyle[\mathcal{M}_{Y},\mathcal{M}_{Z}]=$ $\displaystyle\frac{1}{32\pi
G}\int dud\Omega[\mathcal{M}_{Y},\dot{C}^{BC}\Delta_{BC}(Z;C;u,\Omega)]$
$\displaystyle=$ $\displaystyle\frac{-i}{32\pi G}\int
dud\Omega\Big{[}\big{[}\Delta^{BC}(Y;\dot{C};u,\Omega)+\frac{1}{2}\Delta^{BC}(\dot{Y};C;u,\Omega)\big{]}\Delta_{BC}(Z;C;u,\Omega)$
(E.2)
$\displaystyle+2\dot{C}^{BC}Z^{A}\nabla^{D}\Delta^{EF}(Y;C;u,\Omega)\rho_{AEFDBC}+\dot{C}^{BC}\nabla^{D}Z^{A}\Delta^{EF}(Y;C;u,\Omega)P_{AEFDBC}\Big{]}$
$\displaystyle+\frac{i}{64\pi G}\int dud\Omega
du^{\prime}\alpha(u^{\prime}-u)\dot{C}^{BC}$
$\displaystyle\times\big{[}2\rho_{AEFDBC}Z^{A}\nabla^{D}\Delta^{EF}(\dot{Y};C;u^{\prime},\Omega)+P_{AEFDBC}\nabla^{D}Z^{A}\Delta^{EF}(\dot{Y};C;u^{\prime},\Omega)\big{]}.$

Using the integration by part several times, we can obtain

$\displaystyle[\mathcal{M}_{Y},\mathcal{M}_{Z}]=$
$\displaystyle\frac{-i}{32\pi G}\int
dud\Omega\big{[}\Delta^{EF}(Y;\dot{C})\Delta_{EF}(Z;C)-\Delta^{EF}(Y;C)\Delta_{EF}(Z;\dot{C})\big{]}$
$\displaystyle+\frac{i}{64\pi G}\int dud\Omega
du^{\prime}\alpha(u^{\prime}-u)\Delta_{EF}(\dot{Z};C;u,\Omega)\Delta^{EF}(\dot{Y};C;u^{\prime},\Omega).$
(E.3)

The non-local term is precisely the previous $N_{M}(Y,Z)$, while for local
terms, one can further simplify to get

$\displaystyle\frac{i}{32\pi G}\int
dud\Omega\dot{C}_{EF}\big{[}\Delta^{EF}(Y;\Delta(Z;C))-\Delta^{EF}(Z;\Delta(Y;C))\big{]}.$
(E.4)

To form the local operators, we need use an identity

$\displaystyle\Delta_{EF}(Y;\Delta(Z;C))-\Delta_{EF}(Z;\Delta(Y;C))=\Delta_{EF}([Y,Z];C)+2o(Y,Z)C^{BC}Q_{EFBC},$
(E.5)

whose proof demand some properties of higher rank tensor. The main ones are

$\displaystyle 2\rho_{ABCDEF}P^{GIJHBC}=\gamma_{AD}P^{GIJH}{}_{EF},\qquad
2\rho_{AIJDEF}P^{GIJHBC}=\gamma_{AD}P^{GEFH}{}_{BC},$ (E.6) $\displaystyle
P_{ABCDEF}P^{GIJHBC}-P^{G}{}_{BC}{}^{H}{}_{EF}P_{A}{}^{IJ}{}_{D}{}^{BC}=0,$
(E.7)

and

$\displaystyle
2Q^{ABCD}Q_{EF}{}^{IJ}=\gamma^{CB}P^{DIJA}{}_{EF}-\gamma^{AD}P^{BIJC}{}_{EF}.$
(E.8)

Now it is easy to find the local parts of $[\mathcal{M}_{Y},\mathcal{M}_{Z}]$
can be written as

$\displaystyle\frac{i}{32\pi G}\int
dud\Omega\dot{C}^{EF}\big{[}\Delta_{EF}([Y,Z];C)+2o(Y,Z)C^{IJ}Q_{EFIJ}\big{]}=i\mathcal{M}_{[Y,Z]}+2i\mathcal{O}_{o(Y,Z)}.$
(E.9)

As for central charges, one need start from correlation functions of shear
tensor, and we will not show the details here.

For other commutators, we provide the key identities that may be used. The
following identity is useful for the calculation of
$[\mathcal{T}_{f},\mathcal{M}_{Y}]$

$\displaystyle
P_{ABCDEF}+\epsilon_{DA}Q_{EFBC}-\frac{1}{4}\gamma_{AD}P_{BEFC}=0.$ (E.10)

To calculate $[\mathcal{M}_{Y},\mathcal{O}_{g}]$, one might make use of

$\displaystyle 2Q^{BC}{}_{EF}\rho_{ABCDGH}=\gamma_{AD}Q_{GHEF},$ (E.11)

and

$\displaystyle
P_{ABCD}{}^{EF}Q_{EFIJ}-P_{AIJD}{}^{EF}Q_{EFBC}=\gamma_{AD}Q_{BCIJ}.$ (E.12)

The remaining commutators are relatively straightforward.

##  Appendix F Conserved current for duality transformation

The PF action is not invariant under duality transformation. Just like the
electromagnetic theory, we may construct a duality symmetric action

$\displaystyle S[h,\widetilde{h}]=\frac{1}{2}(S_{\rm PF}[h]+S_{\rm
PF}[\widetilde{h}]).$ (F.1)

One can derive the equations of motion from this symmetric action. More
importantly, the action is invariant under duality transformation. To prove
this, we note that the infinitesimal duality transformation is

$\displaystyle\delta_{\epsilon}h_{\mu\nu}=\epsilon\widetilde{h}_{\mu\nu},\qquad\delta_{\epsilon}\widetilde{h}_{\mu\nu}=-\epsilon
h_{\mu\nu}.$ (F.2)

Therefore, the variation of the symmetric action is

$\displaystyle\delta_{\epsilon}S[h,\widetilde{h}]$ $\displaystyle=$
$\displaystyle-\frac{1}{64\pi G}\int
d^{4}xL^{\mu_{1}\mu_{2}\cdots\mu_{6}}\partial_{\mu_{1}}h_{\mu_{2}\mu_{3}}\delta_{\epsilon}\partial_{\mu_{4}}h_{\mu_{5}\mu_{6}}-\frac{1}{64\pi
G}\int
d^{4}xL^{\mu_{1}\mu_{2}\cdots\mu_{6}}\partial_{\mu_{1}}\widetilde{h}_{\mu_{2}\mu_{3}}\delta_{\epsilon}\partial_{\mu_{4}}\widetilde{h}_{\mu_{5}\mu_{6}}$
(F.3) $\displaystyle=$ $\displaystyle-\frac{\epsilon}{64\pi G}\int
d^{4}xL^{\mu_{1}\mu_{2}\cdots\mu_{6}}\partial_{\mu_{1}}h_{\mu_{2}\mu_{3}}\partial_{\mu_{4}}\widetilde{h}_{\mu_{5}\mu_{6}}+\frac{\epsilon}{64\pi
G}\int
d^{4}xL^{\mu_{1}\mu_{2}\cdots\mu_{6}}\partial_{\mu_{1}}\widetilde{h}_{\mu_{2}\mu_{3}}\partial_{\mu_{4}}h_{\mu_{5}\mu_{6}}$
$\displaystyle=$ $\displaystyle 0.$

At the last step, we used the fact that the tensor
$L^{\mu_{1}\mu_{2}\cdots\mu_{6}}$ is invariant under the exchange of indices

$\displaystyle\mu_{1}\mu_{2}\mu_{3}\leftrightarrow\mu_{4}\mu_{5}\mu_{6}.$
(F.4)

Now using the Noether’s theorem, the conserved current is

$\displaystyle j_{\text{duality}}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial\mathcal{L}_{\rm
PF}(h)}{\partial(\partial_{\mu}h_{\rho\sigma})}\delta
h_{\rho\sigma}+\frac{1}{2}\frac{\partial\mathcal{L}_{\rm
PF}(\widetilde{h})}{\partial(\partial_{\mu}\widetilde{h}_{\rho\sigma})}\delta\widetilde{h}_{\rho\sigma}$
(F.5) $\displaystyle=$ $\displaystyle\frac{1}{64\pi
G}L^{\mu\rho\sigma\mu_{4}\mu_{5}\mu_{6}}(h_{\rho\sigma}\partial_{\mu_{4}}\widetilde{h}_{\mu_{5}\mu_{6}}-\widetilde{h}_{\rho\sigma}\partial_{\mu_{4}}h_{\mu_{5}\mu_{6}}).$

At the first line, $\mathcal{L}_{\rm PF}(h)$ is the Lagrangian density

$\displaystyle\mathcal{L}_{\rm PF}(h)=-\frac{1}{64\pi
G}L^{\mu_{1}\mu_{2}\cdots\mu_{6}}\partial_{\mu_{1}}h_{\mu_{2}\mu_{3}}\partial_{\mu_{4}}h_{\mu_{5}\mu_{6}}.$
(F.6)

At the second line, we have discarded the constant $\epsilon$. One can use the
equations of motion to prove the conservation

$\displaystyle\partial_{\mu}j_{\text{duality}}^{\mu}=0.$ (F.7)

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