<p>We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When <i>A</i> is a rooted directed tree, we show that the set of hypercyclic vectors of any weighted backward shift operator on the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$c_0(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\ell ^1(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell ^p(A), 1<p<+\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing weighted backward shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.</p>

Hypercyclic algebras for weighted shifts on trees

期刊论文
发表日期：

2026年5月12日

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Arafat Abbar,
Fernando Costa Jr.

摘要

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When A is a rooted directed tree, we show that the set of hypercyclic vectors of any weighted backward shift operator on the space $c_0(A)$ $c_{0} (A)$ or $\ell ^1(A)$ $ℓ^{1} (A)$ is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on $\ell ^p(A), 1<p<+\infty $ $ℓ^{p} (A), 1 < p < + \infty$ . Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing weighted backward shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.