<p>The nonlinear Moore–Penrose metric generalized inverse has been introduced as a tool for analyzing the best approximate solutions to ill-posed operator equations in Banach spaces. Building on recent advancements, this paper mainly investigates the representations and reverse order laws for the Moore–Penrose metric generalized inverse of Banach space operator. Specifically, by exploiting certain geometric properties of Banach spaces and the characteristics of the metric projection, and under the appropriate assumption of quasi-additivity, we derive various representations of the Moore–Penrose metric generalized inverse in terms of the ranges and null spaces of operators. Furthermore, we explore the reverse order laws for the Moore–Penrose metric generalized inverse under compositions of bounded linear operators. In particular, we demonstrate that for two bounded linear operators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2950_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ A $</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2950_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ B $</EquationSource> <EquationSource Format="MATHML"><math> <mi>B</mi> </math></EquationSource> </InlineEquation> acting on Banach spaces, the reverse order law <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2950_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">$ (AB)^M = B^M A^M $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mi>M</mi> </msup> <mo>=</mo> <msup> <mi>B</mi> <mi>M</mi> </msup> <msup> <mi>A</mi> <mi>M</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> holds under suitable conditions. These findings provide new insights into the structure of nonlinear generalized inverses of Banach space operators.</p>

Representations and Reverse Order Laws for the Nonlinear Moore–Penrose Metric Generalized Inverse of Banach Space Operator

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发表日期：

2025年9月22日

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Jianbing Cao

摘要

The nonlinear Moore–Penrose metric generalized inverse has been introduced as a tool for analyzing the best approximate solutions to ill-posed operator equations in Banach spaces. Building on recent advancements, this paper mainly investigates the representations and reverse order laws for the Moore–Penrose metric generalized inverse of Banach space operator. Specifically, by exploiting certain geometric properties of Banach spaces and the characteristics of the metric projection, and under the appropriate assumption of quasi-additivity, we derive various representations of the Moore–Penrose metric generalized inverse in terms of the ranges and null spaces of operators. Furthermore, we explore the reverse order laws for the Moore–Penrose metric generalized inverse under compositions of bounded linear operators. In particular, we demonstrate that for two bounded linear operators $ A $ $A$ and $ B $ $B$ acting on Banach spaces, the reverse order law $ (AB)^M = B^M A^M $ ${(A B)}^{M} = B^{M} A^{M}$ holds under suitable conditions. These findings provide new insights into the structure of nonlinear generalized inverses of Banach space operators.