<p>We give the extension theorem for increasing order continuous orthogonallyadditive operators which is similar to Veksler’s theorem for order continuouspositive linear operators. We also show that an orthogonally additiveoperator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1527_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$ T $</EquationSource> </InlineEquation> from a Riesz space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1527_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ E $</EquationSource> </InlineEquation> with principal projection property toa Riesz space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1527_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$ F $</EquationSource> </InlineEquation>, is order continuous if and only if for every <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1527_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$ x\in E $</EquationSource> </InlineEquation> therestriction of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1527_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$ T $</EquationSource> </InlineEquation> to the ideal generated by <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1527_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ x $</EquationSource> </InlineEquation> is order continuous.</p>

Extension and Restriction of Orthogonally Additive Operators

期刊论文
发表日期：

2025年1月29日

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D. Tulu,
B. Turan

摘要

We give the extension theorem for increasing order continuous orthogonallyadditive operators which is similar to Veksler’s theorem for order continuouspositive linear operators. We also show that an orthogonally additiveoperator $ T $ from a Riesz space $ E $ with principal projection property toa Riesz space $ F $ , is order continuous if and only if for every $ x\in E $ therestriction of $ T $ to the ideal generated by $ x $ is order continuous.