Let X be a Banach space of dimension greater than 2. We show that if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$ \varphi : B(X) \rightarrow B(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a linear map satisfying <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="242" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi ([A, B])= [\varphi (A), B] + [A, \varphi (B)]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$ A, B \in B(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$ AB=F $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation>, where F is an arbitrary but fixed operator satisfying there exists some nontrivial idempotent P such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$ F=PF $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>=</mo> <mi>P</mi> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation> (resp., <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$ F=FP $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>=</mo> <mi>F</mi> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation>), then <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="186" /> </InlineMediaObject> <EquationSource Format="TEX">$ \varphi (A) = AT-TA+\gamma (A) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>T</mi> <mo>-</mo> <mi>T</mi> <mi>A</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$ A \in B(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$ T \in B(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$ \gamma : B(X) \rightarrow {\mathbb {C}}I $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">C</mi> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> is a linear map vanishing on each commutator [A, B] whenever <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1712_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$ AB=F $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation>. Those main results of Lu and Jing [F. Lu, W. Jing, Characterizations of Lie derivations of B(X), Linear Algebra Appl. 432(2010), 89-99] are improved.

Lie Derivations of Operator Algebras on Banach Spaces

期刊论文
发表日期：

2025年5月22日

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Lei Liu,
Suqian Hou

摘要

Let X be a Banach space of dimension greater than 2. We show that if $ \varphi : B(X) \rightarrow B(X) $ $φ : B (X) \to B (X)$ is a linear map satisfying $\varphi ([A, B])= [\varphi (A), B] + [A, \varphi (B)]$ $φ ([A, B]) = [φ (A), B] + [A, φ (B)]$ for any $ A, B \in B(X) $ $A, B \in B (X)$ with $ AB=F $ $A B = F$ , where F is an arbitrary but fixed operator satisfying there exists some nontrivial idempotent P such that $ F=PF $ $F = P F$ (resp., $ F=FP $ $F = F P$ ), then $ \varphi (A) = AT-TA+\gamma (A) $ $φ (A) = A T - T A + γ (A)$ for all $ A \in B(X) $ $A \in B (X)$ , where $ T \in B(X) $ $T \in B (X)$ and $ \gamma : B(X) \rightarrow {\mathbb {C}}I $ $γ : B (X) \to C I$ is a linear map vanishing on each commutator [A, B] whenever $ AB=F $ $A B = F$ . Those main results of Lu and Jing [F. Lu, W. Jing, Characterizations of Lie derivations of B(X), Linear Algebra Appl. 432(2010), 89-99] are improved.