<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the algebra of all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> complex matrices. For any matrix <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A\in \mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A^{\dagger }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mo>†</mo> </msup> </math></EquationSource> </InlineEquation> denote the adjoint and Moore–Penrose inverse of <i>A</i>, respectively. For a positive scalar <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, the Bourhim–Mbekhta transformation <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Theta _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> is defined on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by <Equation ID="Equ26"> <EquationSource Format="TEX">$ \Theta _\alpha (A):= \frac{1}{\alpha +1}\left( \alpha A + A^{\dagger }\right) , \quad (A\in \mathscr {M}_n(\mathbb {C})). $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mfenced close=")" open="("> <mi>α</mi> <mi>A</mi> <mo>+</mo> <msup> <mi>A</mi> <mrow> <mrow /> <mo>∗</mo> <mo>†</mo> </mrow> </msup> </mfenced> <mo>,</mo> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>In this paper, we show that if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> is a map on <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Theta _\alpha \!\left( \phi (A)\phi (B)\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Theta _\alpha (AB)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are unitarily similar for all <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$A,B\in \mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if and only if there exist a unitary matrix <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$U\in \mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and a scalar <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\epsilon ^2=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ϵ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, such that <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\phi (A)=\epsilon UAU^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>ϵ</mi> <mi>U</mi> <mi>A</mi> <msup> <mi>U</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$A\in \mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also characterize all maps <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$\Theta _\alpha (ABA)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$\Theta _\alpha (\phi (A)\phi (B)\phi (A))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are unitarily similar for every <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$A,B\in \mathscr {M}_n(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Unitary Similarity and Preservers of Bourhim–Mbekhta’s Transformation

期刊论文
发表日期：

2026年4月20日

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Mohamed Mabrouk

摘要

Let $n\ge 3$ $n \geq 3$ and let $\mathscr {M}_n(\mathbb {C})$ $M_{n} (C)$ denote the algebra of all $n\times n$ $n \times n$ complex matrices. For any matrix $A\in \mathscr {M}_n(\mathbb {C})$ $A \in M_{n} (C)$ , let $A^*$ $A^{*}$ and $A^{\dagger }$ $A^{†}$ denote the adjoint and Moore–Penrose inverse of A, respectively. For a positive scalar $\alpha $ $α$ , the Bourhim–Mbekhta transformation $\Theta _\alpha $ $Θ_{α}$ is defined on $\mathscr {M}_n(\mathbb {C})$ $M_{n} (C)$ by $ \Theta _\alpha (A):= \frac{1}{\alpha +1}\left( \alpha A + A^{*\dagger }\right) , \quad (A\in \mathscr {M}_n(\mathbb {C})). $ $Θ_{α} (A) : = \frac{1}{α + 1} (α A + A^{* †}), (A \in M_{n} (C)) .$ In this paper, we show that if $\phi $ $ϕ$ is a map on $\mathscr {M}_n(\mathbb {C})$ $M_{n} (C)$ , then $\Theta _\alpha \!\left( \phi (A)\phi (B)\right) $ $Θ_{α} (ϕ (A) ϕ (B))$ and $\Theta _\alpha (AB)$ $Θ_{α} (A B)$ are unitarily similar for all $A,B\in \mathscr {M}_n(\mathbb {C})$ $A, B \in M_{n} (C)$ if and only if there exist a unitary matrix $U\in \mathscr {M}_n(\mathbb {C})$ $U \in M_{n} (C)$ and a scalar $\epsilon $ $ϵ$ with $\epsilon ^2=1$ $ϵ^{2} = 1$ , such that $\phi (A)=\epsilon UAU^*$ $ϕ (A) = ϵ U A U^{*}$ for all $A\in \mathscr {M}_n(\mathbb {C})$ $A \in M_{n} (C)$ . We also characterize all maps $\phi $ $ϕ$ on $\mathscr {M}_n(\mathbb {C})$ $M_{n} (C)$ for which $\Theta _\alpha (ABA)$ $Θ_{α} (A B A)$ and $\Theta _\alpha (\phi (A)\phi (B)\phi (A))$ $Θ_{α} (ϕ (A) ϕ (B) ϕ (A))$ are unitarily similar for every $A,B\in \mathscr {M}_n(\mathbb {C})$ $A, B \in M_{n} (C)$ .