<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> be a generalized matrix algebra. A linear map <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi : \mathcal {G} \rightarrow \mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">G</mi> </mrow> </math></EquationSource> </InlineEquation> is said to be an anti-derivation at zero if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">$T\varphi (S)+\varphi (T)S=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mi>S</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$S, T \in \mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>,</mo> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">G</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$ST=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we describe the general form of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> and consider the question of when <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2024_1814_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> equals to zero. The results are then applied to full matrix algebras and some operator algebras.</p>

Characterization of Anti-derivations on Generalized Matrix Algebras

期刊论文
发表日期：

2025年1月6日

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Lei Liu,
Suqian Hou

摘要

Let $\mathcal {G}$ $G$ be a generalized matrix algebra. A linear map $\varphi : \mathcal {G} \rightarrow \mathcal {G}$ $φ : G \to G$ is said to be an anti-derivation at zero if $T\varphi (S)+\varphi (T)S=0$ $T φ (S) + φ (T) S = 0$ for every $S, T \in \mathcal {G}$ $S, T \in G$ with $ST=0$ $S T = 0$ . In this paper, we describe the general form of $\varphi $ $φ$ and consider the question of when $\varphi $ $φ$ equals to zero. The results are then applied to full matrix algebras and some operator algebras.