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<p>In this paper, Roth-type solvability criteria are proposed for the generalized Sylvester equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_452_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="202" /> </InlineMediaObject> <EquationSource Format="TEX">\(AXB+CYD+EZF=G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mi>Y</mi> <mi>D</mi> <mo>+</mo> <mi>E</mi> <mi>Z</mi> <mi>F</mi> <mo>=</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> on finite dimensional spaces and infinite dimensional Hilbert spaces, respectively. Moreover, we give a solvability condition for the operator equation <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_452_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">\(AXB-CXD = E\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>-</mo> <mi>C</mi> <mi>X</mi> <mi>D</mi> <mo>=</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> by only using one invertible operator.</p>

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On Roth-type solvability criteria for generalized Sylvester matrix and operator equations

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Hua Wang,
  • Xiaoju Jin,
  • Junjie Huang

摘要

In this paper, Roth-type solvability criteria are proposed for the generalized Sylvester equation \(AXB+CYD+EZF=G\) A X B + C Y D + E Z F = G on finite dimensional spaces and infinite dimensional Hilbert spaces, respectively. Moreover, we give a solvability condition for the operator equation \(AXB-CXD = E\) A X B - C X D = E by only using one invertible operator.