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<p>For given linear operators <i>A</i>, <i>B</i>, and <i>C</i>, we present a new characterization of left and right (<i>B</i>,&#xa0;<i>C</i>)-invertible operators through specific projections in the Banach operators space. We define a wider class of polarity, called one sided (<i>B</i>,&#xa0;<i>C</i>)-polarity, and establish the equivalence between these concepts under the regularity of <i>CAB</i>. As a fundamental result, we derive the explicit formulas of left and right (<i>B</i>,&#xa0;<i>C</i>)-inverses via these projections. Furthermore, we provide a new characterization of several well-known generalized inverses, with particular focus on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1231_Article_IEq1.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\( A^{(2)}_{\mathcal {T,S}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mrow> <mi mathvariant="script">T</mi> <mo>,</mo> <mi mathvariant="script">S</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>.</p>

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One sided (B, C)-polarity in Banach space operators

  • 期刊论文

  • 发表日期：

    2025年5月5日

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  • Btissam Laghmam,
  • Hassane Zguitti

摘要

For given linear operators A, B, and C, we present a new characterization of left and right (B, C)-invertible operators through specific projections in the Banach operators space. We define a wider class of polarity, called one sided (B, C)-polarity, and establish the equivalence between these concepts under the regularity of CAB. As a fundamental result, we derive the explicit formulas of left and right (B, C)-inverses via these projections. Furthermore, we provide a new characterization of several well-known generalized inverses, with particular focus on \( A^{(2)}_{\mathcal {T,S}}\) A T , S ( 2 ) .