<p>Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">E</mi> </math></EquationSource> </InlineEquation> be a unital <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq2.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-algebra and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta \ne -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>≠</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be a nonzero scalar. This paper establishes that if a nonlinear mapping <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta : \mathfrak {E} \rightarrow \mathfrak {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo>:</mo> <mi mathvariant="fraktur">E</mi> <mo stretchy="false">→</mo> <mi mathvariant="fraktur">E</mi> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="477" /> </InlineMediaObject> <EquationSource Format="TEX">$ \zeta (\mathcal {U} \bullet \mathcal {V} \circ _\eta \mathcal {W})=\zeta (\mathcal {U}) \bullet \mathcal {V} \circ _\eta \mathcal {W}+\mathcal {U} \bullet \zeta (\mathcal {V}) \circ _\eta \mathcal {W}+\mathcal {U} \bullet \mathcal {V} \circ _\eta \zeta (\mathcal {W})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">U</mi> <mo>∙</mo> <mi mathvariant="script">V</mi> <msub> <mo>∘</mo> <mi>η</mi> </msub> <mi mathvariant="script">W</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">U</mi> <mo stretchy="false">)</mo> </mrow> <mo>∙</mo> <mi mathvariant="script">V</mi> <msub> <mo>∘</mo> <mi>η</mi> </msub> <mi mathvariant="script">W</mi> <mo>+</mo> <mi mathvariant="script">U</mi> <mo>∙</mo> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">V</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>∘</mo> <mi>η</mi> </msub> <mi mathvariant="script">W</mi> <mo>+</mo> <mi mathvariant="script">U</mi> <mo>∙</mo> <mi mathvariant="script">V</mi> <msub> <mo>∘</mo> <mi>η</mi> </msub> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">W</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathcal {U}, \mathcal {V}, \mathcal {W} \in \mathfrak {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo>,</mo> <mi mathvariant="script">V</mi> <mo>,</mo> <mi mathvariant="script">W</mi> <mo>∈</mo> <mi mathvariant="fraktur">E</mi> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq12.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> is an additive <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq2.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-derivation and fulfils <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq14.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta (\eta \mathcal {U})=\eta \zeta (\mathcal {U})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>η</mi> <mi mathvariant="script">U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>η</mi> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">U</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_608_Article_IEq15.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {U} \in \mathfrak {E}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo>∈</mo> <mi mathvariant="fraktur">E</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Additionally, we extend this result to various other algebras.</p>

Nonlinear mixed skew and $\eta $-Jordan derivations on $*$-algebras

期刊论文
发表日期：

2025年8月26日

查看全文

Asma Ali,
Shakiv Ali,
Mohd Tasleem

摘要

Let $\mathfrak {E}$ $E$ be a unital $*$ $*$ -algebra and $\eta \ne -1$ $η \neq - 1$ be a nonzero scalar. This paper establishes that if a nonlinear mapping $\zeta : \mathfrak {E} \rightarrow \mathfrak {E}$ $ζ : E \to E$ satisfies $ \zeta (\mathcal {U} \bullet \mathcal {V} \circ _\eta \mathcal {W})=\zeta (\mathcal {U}) \bullet \mathcal {V} \circ _\eta \mathcal {W}+\mathcal {U} \bullet \zeta (\mathcal {V}) \circ _\eta \mathcal {W}+\mathcal {U} \bullet \mathcal {V} \circ _\eta \zeta (\mathcal {W})$ $ζ (U ∙ V \circ_{η} W) = ζ (U) ∙ V \circ_{η} W + U ∙ ζ (V) \circ_{η} W + U ∙ V \circ_{η} ζ (W)$ for all $ \mathcal {U}, \mathcal {V}, \mathcal {W} \in \mathfrak {E}$ $U, V, W \in E$ , then $\zeta $ $ζ$ is an additive $*$ $*$ -derivation and fulfils $\zeta (\eta \mathcal {U})=\eta \zeta (\mathcal {U})$ $ζ (η U) = η ζ (U)$ for all $\mathcal {U} \in \mathfrak {E}.$ $U \in E .$ Additionally, we extend this result to various other algebras.

Nonlinear mixed skew and \(\eta \)-Jordan derivations on \(*\)-algebras

摘要