Algebrability of the set of hypercyclic vectors for backward shift operators

Falcó, Javier; Grosse‐Erdmann, Karl‐G.

doi:10.1016/j.aim.2020.107082

Cited by 10 publications

(10 citation statements)

References 11 publications

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“…Remark We note that the result holds in even greater generality. It remains true, with essentially the same proof, for any weighted backward shift

B_{w}

on any Fréchet sequence algebra X under coordinatewise multiplication provided only that the (unweighted) backward shift B is an operator on X so that

B^{n} x \to 0

for all

x \in X

; see [16] for the relevant notions.…”

Section: The Rolewicz Operator and Scripta‐hypercyclicitymentioning

confidence: 98%

“…It is proved in [16] that in the present setting the set of hypercyclic vectors for the Rolewicz operator is algebrable. As a consequence of the proposition we see that no hypercyclic algebra for the Rolewicz operator can contain a frequently hypercyclic vector.…”

Section: The Rolewicz Operator and The Coordinatewise Multiplicative mentioning

confidence: 99%

“…It is well known that D is frequently hypercyclic, see [6]. When we identify

H (C)

with the space of sequences of Taylor coefficients at 0 of entire functions, D becomes a weighted backward shift with weights

w_{n} = n

n \geq 1

D (x (0), x (1), x (2), \dots) = (x (1), 2 x (2), 3 x (3), \dots), x \in H (C) .

We turn

H (C)

into a Fréchet algebra by endowing it with the product of coordinatewise multiplication, also called the Hadamard product on

H (C)

; see [16] for more details.…”

Section: The Maclane Operator and Frequently Hypercyclic Algebrasmentioning

confidence: 99%

“…Motivated by the result for the MacLane differentiation operator obtained in [2], Shkarin [20] and Bayart and Matheron [7, Theorem 8.26] showed independently that it admits a hypercyclic algebra, thereby providing the first known example of an operator with a hypercyclic algebra. Recently, Bès, Conejero and Papathanasiou [9, 10, 12] and Bayart [4] extended the existence of hypercyclic algebras to various other operators like convolution operators and certain translation operators, and the authors [16] proved the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product.…”

mentioning

confidence: 99%

“…Here we are interested in the study of the more restrictive case of frequent hypercyclicity; in continuation of our approach in [16], our methods in this paper will be constructive. A (continuous, linear) operator T is called frequently hypercyclic if there is some

x \in X

so that, for any non‐empty open subset U of X ,

\underset{̲}{dens} {n \in {double-struckN}_{0} : T^{n} x \in U} > 0,

where the lower density of a subset

A \subset {double-struckN}_{0}

is defined as

\underset{̲}{dens} false(A false) = \underset{N \to \infty}{lim inf} \frac{card {0 \leq n \leq N : n \in A}}{N + 1} .

In this case, x is called a frequently hypercyclic vector for T .…”

mentioning

confidence: 99%

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Algebras of frequently hypercyclic vectors

Falcó

Grosse‐Erdmann

2020

Mathematische Nachrichten

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We show that the multiples of the backward shift operator on the spaces , 1 ≤ < ∞, or 0 , when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of -hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator on any Banach space, ( ) is algebrable for a suitable product, and in some cases it is even strongly algebrable. K E Y W O R D S-hypercyclic vector, algebrability, Banach algebra, frequently hypercyclic vector, weighted shift 1120

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“…Remark We note that the result holds in even greater generality. It remains true, with essentially the same proof, for any weighted backward shift

B_{w}

on any Fréchet sequence algebra X under coordinatewise multiplication provided only that the (unweighted) backward shift B is an operator on X so that