Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the algebra of all bounded linear operators on a complex Hilbert space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. For an operator <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$T\in \mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$W_0(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the maximal numerical range of T. We show that a map <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> from <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> onto itself satisfies <Equation ID="Equ12"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_Equ12.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="412" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} W_0\left( \varphi (S)\varphi (T)\varphi (S)\right) ~=~W_0(STS), \qquad (T,~S\in \mathscr {L}(\mathscr {H})), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>W</mi> <mn>0</mn> </msub> <mfenced close=")" open="("> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mfenced> <mspace width="3.33333pt" /> <mo>=</mo> <mspace width="3.33333pt" /> <msub> <mi>W</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mi>T</mi> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mi>S</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>if and only if there are a unitary operator <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$U\in \mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq8.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda \in \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda ^3=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>λ</mi> <mn>3</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and either <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (T)= \lambda UTU^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>U</mi> <mi>T</mi> <msup> <mi>U</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$T\in \mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq12.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (T)= \lambda UT^\top U^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>U</mi> <msup> <mi>T</mi> <mi>⊤</mi> </msup> <msup> <mi>U</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$T\in \mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq14.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$T^\top $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mi>⊤</mi> </msup> </math></EquationSource> </InlineEquation> denotes the transpose of any operator <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$T\in \mathscr {L}(\mathscr {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> relative to a fixed but arbitrary orthonormal base of <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. When the triple product “STS” is replaced by the skew-triple product “<InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq17.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$TS^*T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <msup> <mi>S</mi> <mo>∗</mo> </msup> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>”, we arrive at the same conclusion but with <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2148_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

Maps preserving the maximal numerical range of the triple product of operators

期刊论文
发表日期：

2025年7月21日

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Abdellatif Bourhim,
Mohamed Mabrouk

摘要

Let $\mathscr {L}(\mathscr {H})$ $L (H)$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathscr {H}$ $H$ . For an operator $T\in \mathscr {L}(\mathscr {H})$ $T \in L (H)$ , let $W_0(T)$ $W_{0} (T)$ be the maximal numerical range of T. We show that a map $\varphi $ $φ$ from $\mathscr {L}(\mathscr {H})$ $L (H)$ onto itself satisfies $\begin{aligned} W_0\left( \varphi (S)\varphi (T)\varphi (S)\right) ~=~W_0(STS), \qquad (T,~S\in \mathscr {L}(\mathscr {H})), \end{aligned}$ $\begin{matrix} W_{0} (φ (S) φ (T) φ (S)) = W_{0} (S T S), (T, S \in L (H)), \end{matrix}$ if and only if there are a unitary operator $U\in \mathscr {L}(\mathscr {H})$ $U \in L (H)$ and $\lambda \in \mathbb {C}$ $λ \in C$ such that $\lambda ^3=1$ $λ^{3} = 1$ and either $\varphi (T)= \lambda UTU^*$ $φ (T) = λ U T U^{*}$ for all $T\in \mathscr {L}(\mathscr {H})$ $T \in L (H)$ , or $\varphi (T)= \lambda UT^\top U^*$ $φ (T) = λ U T^{⊤} U^{*}$ for all $T\in \mathscr {L}(\mathscr {H})$ $T \in L (H)$ . Here, $T^\top $ $T^{⊤}$ denotes the transpose of any operator $T\in \mathscr {L}(\mathscr {H})$ $T \in L (H)$ relative to a fixed but arbitrary orthonormal base of $\mathscr {H}$ $H$ . When the triple product “STS” is replaced by the skew-triple product “ $TS^*T$ $T S^{*} T$ ”, we arrive at the same conclusion but with $\lambda =1$ $λ = 1$ .