<p>We characterize the strictly increasing symbols <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi :\mathbb {N}_0\longrightarrow \mathbb {N}_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> <mo stretchy="false">⟶</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> whose composition operators <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> satisfy the Frequent Hypercyclicity Criterion on the little Lipschitz space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}_0(\mathbb {N}_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">L</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. With this result we continue the recent research about this kind of spaces and operators, but our approach relies on establishing a natural isomorphism between the Lipschitz-type spaces over rooted trees and the classical spaces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell ^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Such isomorphism provides an alternative framework that simplifies and allows to improve many previous results about these spaces and the operators defined there.</p>

Frequently Hypercyclic Composition Operators on The Little Lipschitz Space of A Rooted Tree

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2026年3月25日

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Antoni López-Martínez

摘要

We characterize the strictly increasing symbols $\varphi :\mathbb {N}_0\longrightarrow \mathbb {N}_0$ $φ : N_{0} ⟶ N_{0}$ whose composition operators $C_{\varphi }$ $C_{φ}$ satisfy the Frequent Hypercyclicity Criterion on the little Lipschitz space $\mathcal {L}_0(\mathbb {N}_0)$ $L_{0} (N_{0})$ . With this result we continue the recent research about this kind of spaces and operators, but our approach relies on establishing a natural isomorphism between the Lipschitz-type spaces over rooted trees and the classical spaces $\ell ^{\infty }$ $ℓ^{\infty}$ and $c_0$ $c_{0}$ . Such isomorphism provides an alternative framework that simplifies and allows to improve many previous results about these spaces and the operators defined there.