<p>For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha \in \mathbb {R},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we consider the scale of function spaces, namely the Dirichlet-type space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {D}_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> consisting of holomorphic functions on the unit bidisk <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {D}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(z,w)=\sum _{k,l=0}^{\infty }a_{kl}z^kw^l$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">kl</mi> </mrow> </msub> <msup> <mi>z</mi> <mi>k</mi> </msup> <msup> <mi>w</mi> <mi>l</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ22"> <EquationSource Format="TEX">$\begin{aligned} \sum _{k,l=0}^{\infty }(k+l+1)^\alpha |a_{kl}|^2 < \infty . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>α</mi> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">kl</mi> </mrow> </msub> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo><</mo> <mi>∞</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2-z_1-z_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>-</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {D}_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \frac{3}{2} < \alpha \le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {D}_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

Cyclic polynomials in Dirichlet-type spaces of the unit bidisk

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2026年6月29日

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Rajkamal Nailwal,
Aljaž Zalar

摘要

For $\alpha \in \mathbb {R},$ $α \in R,$ we consider the scale of function spaces, namely the Dirichlet-type space $\mathcal {D}_{\alpha }$ $D_{α}$ consisting of holomorphic functions on the unit bidisk $\mathbb {D}^2$ $D^{2}$ , $f(z,w)=\sum _{k,l=0}^{\infty }a_{kl}z^kw^l$ $f (z, w) = \sum_{k, l = 0}^{\infty} a_{kl} z^{k} w^{l}$ such that $\begin{aligned} \sum _{k,l=0}^{\infty }(k+l+1)^\alpha |a_{kl}|^2 < \infty . \end{aligned}$ $\begin{matrix} \sum_{k, l = 0}^{\infty} {(k + l + 1)}^{α} {| a_{kl} |}^{2} < \infty . \end{matrix}$ In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial $2-z_1-z_2$ $2 - z_{1} - z_{2}$ in $\mathcal {D}_\alpha $ $D_{α}$ for $ \frac{3}{2} < \alpha \le 2$ $\frac{3}{2} < α \leq 2$ . We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $\mathcal {D}_\alpha $ $D_{α}$ .