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<p>Assume that <i>A</i>, <i>B</i> and <i>C</i> are three linear relations on certain linear spaces. The main objective of this note is to present some necessary and sufficient conditions for the existence of the solutions <i>X</i> of the operator equation <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1836_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">\(A = BXC\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mi>X</mi> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>. The obtained characterizations extend and complete some known operator factorization results due to R.G. Douglas, Z. Sebestyén and D. Popovici.</p>

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Operator Solutions of the Multi–valued Operator Equation \(A = BXC\)

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  • 发表日期：

    2025年2月20日

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  • Adrian Sandovici

摘要

Assume that A, B and C are three linear relations on certain linear spaces. The main objective of this note is to present some necessary and sufficient conditions for the existence of the solutions X of the operator equation \(A = BXC\) A = B X C . The obtained characterizations extend and complete some known operator factorization results due to R.G. Douglas, Z. Sebestyén and D. Popovici.