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<p>We study a class of commuting tuples of bounded linear operators on a complex Hilbert space. This framework extends several well-known classes of operators, including symmetric, <i>A</i>-symmetric, and <i>m</i>-symmetric operators, to a multivariable setting. We begin by formulating the notion of (<i>A</i>,&#xa0;<i>m</i>)-symmetry for tuples and analyze the structural properties such tuples must satisfy. We provide new algebraic and analytic characterizations for such operator tuples. Next, we show that if <i>m</i> is even, then any (<i>A</i>,&#xa0;<i>m</i>)-symmetric tuple is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1284_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\((A,m-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-symmetric tuple. Further, we explore stability under perturbations by <i>q</i>-nilpotent operator. We also study the sum of two (<i>A</i>,&#xa0;<i>m</i>)-symmetric tuple and we give basic properties that generalizes those given in Chō et al. (Adv. Oper. Theory 2:468—474, 2017), Chō et al. (Mediterr. J. Math. 13:2025—2038, 2016), Chō et al. (Stud. Univ. Babeş-Bolyai Math. 62:233—248, 2017), Gu and Stankus (Linear Algebra Appl. 469:500—509, 2015), Jeridi and Rabaoui (Results Math. 74:1—33, 2019).</p>

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Some results on higher order (A, m)-symmetric commuting tuples of operators on a Hilbert space

  • 期刊论文

  • 发表日期：

    2025年7月29日

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  • Nader Jeridi

摘要

We study a class of commuting tuples of bounded linear operators on a complex Hilbert space. This framework extends several well-known classes of operators, including symmetric, A-symmetric, and m-symmetric operators, to a multivariable setting. We begin by formulating the notion of (A, m)-symmetry for tuples and analyze the structural properties such tuples must satisfy. We provide new algebraic and analytic characterizations for such operator tuples. Next, we show that if m is even, then any (A, m)-symmetric tuple is \((A,m-1)\) ( A , m - 1 ) -symmetric tuple. Further, we explore stability under perturbations by q-nilpotent operator. We also study the sum of two (A, m)-symmetric tuple and we give basic properties that generalizes those given in Chō et al. (Adv. Oper. Theory 2:468—474, 2017), Chō et al. (Mediterr. J. Math. 13:2025—2038, 2016), Chō et al. (Stud. Univ. Babeş-Bolyai Math. 62:233—248, 2017), Gu and Stankus (Linear Algebra Appl. 469:500—509, 2015), Jeridi and Rabaoui (Results Math. 74:1—33, 2019).