<p>Proceeding through a representation theoretic approach, we study the family of Toeplitz operators on the weighted Bergman space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}^2_{\alpha }(\mathbb {B}_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mi>α</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> whose symbols are invariant under the action of the semidirect group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S_{\Delta } \rtimes \mathbb {T}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">Δ</mi> </msub> <mo>⋊</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S_{\Delta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi mathvariant="normal">Δ</mi> </msub> </math></EquationSource> </InlineEquation> is a subgroup of permutations. Associating a suitable partition <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> we associated the conjugation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$J_{A}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> on the Bergman space. Moreover we characterize the complex symmetric Toeplitz operators <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$T_{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T_{a(r)t^{p}\overline{t}^{q}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>t</mi> <mi>p</mi> </msup> <msup> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>q</mi> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> with respect the conjugation <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$J_A$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation>, where the symbols are invariant under the action of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$S_{\Delta } \rtimes \mathbb {T}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">Δ</mi> </msub> <mo>⋊</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

Complex symmetric Toeplitz operators through representation theory tools

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2026年6月22日

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Esteban Heredia-Muñoz,
Armando Sánchez-Nungaray,
Carlos-Hugo Jiménez-Gómez

摘要

Proceeding through a representation theoretic approach, we study the family of Toeplitz operators on the weighted Bergman space $\mathcal {H}^2_{\alpha }(\mathbb {B}_n)$ $H_{α}^{2} (B_{n})$ whose symbols are invariant under the action of the semidirect group $S_{\Delta } \rtimes \mathbb {T}^{n}$ $S_{Δ} ⋊ T^{n}$ , where $S_{\Delta }$ $S_{Δ}$ is a subgroup of permutations. Associating a suitable partition $\Delta $ $Δ$ we associated the conjugation $J_{A}$ $J_{A}$ on the Bergman space. Moreover we characterize the complex symmetric Toeplitz operators $T_{a}$ $T_{a}$ and $T_{a(r)t^{p}\overline{t}^{q}}$ $T_{a (r) t^{p} {\bar{t}}^{q}}$ with respect the conjugation $J_A$ $J_{A}$ , where the symbols are invariant under the action of $S_{\Delta } \rtimes \mathbb {T}^{n}$ $S_{Δ} ⋊ T^{n}$ .