Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> be a complex Hilbert space of dimension at least 3 and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the algebra of all bounded linear operators on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. For any <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$A, B\in {\mathcal {B}}({\mathcal {H}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, A and B are said to be orthogonal if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$A^*B=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mo>∗</mo> </msup> <mi>B</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we establish the general form of orthogonality preserving bijections on <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we obtain a characterization of bijections <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi :\mathcal {B(H)}\rightarrow \mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (A)\bot (\varphi (B)-\varphi (C))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>⊥</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$A\bot (B-C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>⊥</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo>-</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_406_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">$A,B,C\in \mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>∈</mo> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Maps preserving certain orthogonality of operators on $\mathcal {B(H)}$

期刊论文
发表日期：

2025年2月7日

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Jingzhou Han,
Weijuan Shi,
Guoxing Ji

摘要

Let ${\mathcal {H}}$ $H$ be a complex Hilbert space of dimension at least 3 and $\mathcal {B(H)}$ $B (H)$ the algebra of all bounded linear operators on ${\mathcal {H}}$ $H$ . For any $A, B\in {\mathcal {B}}({\mathcal {H}})$ $A, B \in B (H)$ , A and B are said to be orthogonal if $A^*B=0$ $A^{*} B = 0$ . In this paper, we establish the general form of orthogonality preserving bijections on $\mathcal {B(H)}$ $B (H)$ . Furthermore, we obtain a characterization of bijections $\varphi :\mathcal {B(H)}\rightarrow \mathcal {B(H)}$ $φ : B (H) \to B (H)$ satisfying $\varphi (A)\bot (\varphi (B)-\varphi (C))$ $φ (A) ⊥ (φ (B) - φ (C))$ if and only if $A\bot (B-C)$ $A ⊥ (B - C)$ for any $A,B,C\in \mathcal {B(H)}$ $A, B, C \in B (H)$ .

Maps preserving certain orthogonality of operators on \(\mathcal {B(H)}\)

摘要