In this paper, motivated by recent developments in linear dynamics, we provide several sufficient conditions for weighted backward shifts to admit an upper frequently hypercyclic algebra on the sequence spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell ^p(A)$</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> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <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> over a rooted directed tree (A, E), where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$1 \le p < +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. To illustrate these criteria, we construct concrete examples of weighted backward shifts satisfying these conditions. Furthermore, we establish a necessary condition for a weighted backward shift to support a frequently hypercyclic algebra on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell ^p(A)$</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> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1 \le p < +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.

Upper frequently hypercyclic algebras for backward shifts on directed trees

期刊论文
发表日期：

2026年7月6日

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Xiang Chen,
Li Zhang,
Zehua Zhou

摘要

In this paper, motivated by recent developments in linear dynamics, we provide several sufficient conditions for weighted backward shifts to admit an upper frequently hypercyclic algebra on the sequence spaces $\ell ^p(A)$ $ℓ^{p} (A)$ and $c_0(A)$ $c_{0} (A)$ over a rooted directed tree (A, E), where $1 \le p < +\infty $ $1 \leq p < + \infty$ . To illustrate these criteria, we construct concrete examples of weighted backward shifts satisfying these conditions. Furthermore, we establish a necessary condition for a weighted backward shift to support a frequently hypercyclic algebra on $\ell ^p(A)$ $ℓ^{p} (A)$ for $1 \le p < +\infty $ $1 \leq p < + \infty$ .