Let E and F be two Banach spaces. Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$g: E\rightarrow F$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation> be an <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-phase isometry for some <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (i.e., for all <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$u,v\in E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation><Equation ID="Equ10"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_Equ10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="454" /> </MediaObject> <EquationSource Format="TEX">$|\; |\Vert g(u)+g(v)\Vert \pm \Vert g(u)-g(v)\Vert |-|\Vert u+v\Vert \pm \Vert u-v\Vert |\;|\le \varepsilon ).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">|</mo> <mspace width="0.277778em" /> <mo stretchy="false">|</mo> <mo stretchy="false">‖</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>±</mo> <mo stretchy="false">‖</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> <mo stretchy="false">‖</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">‖</mo> <mo>±</mo> <mo stretchy="false">‖</mo> <mi>u</mi> <mo>-</mo> <mi>v</mi> <mo stretchy="false">‖</mo> <mo stretchy="false">|</mo> <mspace width="0.277778em" /> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </Equation>In particular, g is said a phase isometry if <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Then we study the relationship between <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-phase isometry and phase isometry from E to F. Finally, we study a Hyers–Ulam type problem for <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_793_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-phase isometries.

On $\varepsilon $-phase isometries and phase isometries

期刊论文
发表日期：

2025年5月13日

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Duanxu Dai,
Zhiruo Lin,
Haixin Que,
Longfa Sun

摘要

Let E and F be two Banach spaces. Let $g: E\rightarrow F$ $g : E \to F$ be an $\varepsilon $ $ε$ -phase isometry for some $\varepsilon \ge 0$ $ε \geq 0$ (i.e., for all $u,v\in E$ $u, v \in E$ $|\; |\Vert g(u)+g(v)\Vert \pm \Vert g(u)-g(v)\Vert |-|\Vert u+v\Vert \pm \Vert u-v\Vert |\;|\le \varepsilon ).$ $| | ‖ g (u) + g (v) ‖ \pm ‖ g (u) - g (v) ‖ | - | ‖ u + v ‖ \pm ‖ u - v ‖ | | \leq ε) .$ In particular, g is said a phase isometry if $\varepsilon =0$ $ε = 0$ . Then we study the relationship between $\varepsilon $ $ε$ -phase isometry and phase isometry from E to F. Finally, we study a Hyers–Ulam type problem for $\varepsilon $ $ε$ -phase isometries.

On \(\varepsilon \)-phase isometries and phase isometries

摘要