In recent years, various aspects of the problem related to the generalization of the class of m-isometries of commuting tuples of operators in Hilbert spaces have appeared in the literature. Let us mention, for example, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3288_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(n_{1},\ldots ,n_{p})$</EquationSource> </InlineEquation>-quasi-m-isometric tuples, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3288_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(m,C)$</EquationSource> </InlineEquation>-isometric tuples, toral m-isometries, and toral <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3288_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(m, A)$</EquationSource> </InlineEquation>-isometries. In this paper, we discuss a problem closely related to this one. We introduce the class of toral-<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3288_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(m,C)$</EquationSource> </InlineEquation>-isometries for a tuple of commuting operators related to a given conjugation operator C on a Hilbert space H. We present its basic properties and demonstrate a variety of results that extend some works related to multivariable operators.

Toral $(m,C)$-isometric multivariable operators

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2025年3月28日

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Sid Ahmed Ould Ahmed Mahmoud,
Abdulrahman Obaid Alshammari,
El Moctar Ould Beiba,
Maawiya Ould Sidi,
Hadi Obaid Alshammari,
Sid Ahmed Ould Beinane

摘要

In recent years, various aspects of the problem related to the generalization of the class of m-isometries of commuting tuples of operators in Hilbert spaces have appeared in the literature. Let us mention, for example, $(n_{1}, \dots, n_{p})$ $(n_{1},\ldots ,n_{p})$ -quasi-m-isometric tuples, $(m, C)$ $(m,C)$ -isometric tuples, toral m-isometries, and toral $(m, A)$ $(m, A)$ -isometries. In this paper, we discuss a problem closely related to this one. We introduce the class of toral- $(m, C)$ $(m,C)$ -isometries for a tuple of commuting operators related to a given conjugation operator C on a Hilbert space H. We present its basic properties and demonstrate a variety of results that extend some works related to multivariable operators.

Toral \((m,C)\)-isometric multivariable operators

摘要