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<p>In this paper, we establish that a matrix is Hermitian if it satisfies specific equations involving its powers and adjoint. Some of these equations are <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1842_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">\(T^n=TT^*\cdots T\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>T</mi> <mi>n</mi> </msup> <mo>=</mo> <mi>T</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mo>⋯</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1842_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="169" /> </InlineMediaObject> <EquationSource Format="TEX">\(TT^*\cdots T=T^*T\cdots T^*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mo>⋯</mo> <mi>T</mi> <mo>=</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mi>T</mi> <mo>⋯</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1842_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(T^n=\operatorname {Re}T\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>T</mi> <mi>n</mi> </msup> <mo>=</mo> <mo>Re</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>. Certain results are also presented in infinite-dimensional spaces.</p>

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Some Conditions Implying the Hermitian Property of Matrices and Operators

  • 期刊论文

  • 发表日期：

    2025年10月24日

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  • Mohammed Hichem Mortad

摘要

In this paper, we establish that a matrix is Hermitian if it satisfies specific equations involving its powers and adjoint. Some of these equations are \(T^n=TT^*\cdots T\) T n = T T ∗ ⋯ T , \(TT^*\cdots T=T^*T\cdots T^*\) T T ∗ ⋯ T = T ∗ T ⋯ T ∗ and \(T^n=\operatorname {Re}T\) T n = Re T . Certain results are also presented in infinite-dimensional spaces.