In this paper, we consider a generalization of the complex Helton class to the multivariable setting. Inspired by the work [Al Rwaily, A., Higher order quasi complex Helton class of Hilbert space operators, Filomat 38 (31), 10819–10834 (2024) https:doi.org/10.2298/FIL2431819A], we introduce the complex Helton class of tuples of commuting operators. Given a positive integer m, for a conjugation C on a complex infinite-dimensional Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> and for commuting d-tuples <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{R}=(R_1,\dots ,R_d),\,\textbf{S}=(S_1,\dots ,S_d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">R</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>R</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> <mi mathvariant="bold">S</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>S</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we say that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">S</mi> </math></EquationSource> </InlineEquation> belongs to the complex Helton class of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">R</mi> </math></EquationSource> </InlineEquation> with order m, and we denote this by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{S}\in Helton_{C,m}(\textbf{R}),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">S</mi> <mo>∈</mo> <mi>H</mi> <mi>e</mi> <mi>l</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>n</mi> <mrow> <mi>C</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> if <Equation ID="Equ24"> <EquationSource Format="TEX">$\begin{aligned} \sum \limits _{k=0}^{m}(-1)^{m-k} \;{m\atopwithdelims ()k}\;\Big (\sum \limits _{i=1}^{d}R_i\Big )^{k}C\Big (\sum \limits _{i=1}^{d}S_i\Big )^{m-k}C=0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mspace width="0.277778em" /> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mi>m</mi> <mi>k</mi> </mfrac> </mfenced> <mspace width="0.277778em" /> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </munderover> <msub> <mi>R</mi> <mi>i</mi> </msub> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>k</mi> </msup> <mi>C</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </munderover> <msub> <mi>S</mi> <mi>i</mi> </msub> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In the present work, some basic structural properties of such a class of a commuting tuples are established, especially the single valued extension property (SVEP) property and Bishop’s property <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Then, we prove that the complex Helton class is invariant under nilpotent perturbation. Finally, spectral properties are shown.

On tuples of commuting operators in the complex Helton class

期刊论文
发表日期：

2026年8月1日

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Mohamed Ben Jbir,
Salah Mecheri,
Rchid Rabaoui

摘要

In this paper, we consider a generalization of the complex Helton class to the multivariable setting. Inspired by the work [Al Rwaily, A., Higher order quasi complex Helton class of Hilbert space operators, Filomat 38 (31), 10819–10834 (2024) https:doi.org/10.2298/FIL2431819A], we introduce the complex Helton class of tuples of commuting operators. Given a positive integer m, for a conjugation C on a complex infinite-dimensional Hilbert space $\mathcal {H}$ $H$ and for commuting d-tuples $\textbf{R}=(R_1,\dots ,R_d),\,\textbf{S}=(S_1,\dots ,S_d)$ $R = (R_{1}, \dots, R_{d}), S = (S_{1}, \dots, S_{d})$ , we say that $\textbf{S}$ $S$ belongs to the complex Helton class of $\textbf{R}$ $R$ with order m, and we denote this by $\textbf{S}\in Helton_{C,m}(\textbf{R}),$ $S \in H e l t o n_{C, m} (R),$ if $\begin{aligned} \sum \limits _{k=0}^{m}(-1)^{m-k} \;{m\atopwithdelims ()k}\;\Big (\sum \limits _{i=1}^{d}R_i\Big )^{k}C\Big (\sum \limits _{i=1}^{d}S_i\Big )^{m-k}C=0. \end{aligned}$ $\begin{matrix} \sum_{k = 0}^{m} {(- 1)}^{m - k} (\binom{m}{k}) (\sum_{i = 1}^{d} R_{i})^{k} C (\sum_{i = 1}^{d} S_{i})^{m - k} C = 0 . \end{matrix}$ In the present work, some basic structural properties of such a class of a commuting tuples are established, especially the single valued extension property (SVEP) property and Bishop’s property $(\beta )$ $(β)$ . Then, we prove that the complex Helton class is invariant under nilpotent perturbation. Finally, spectral properties are shown.