{"title":"Solvability and uniqueness of solution of generalized \u0000 \u0000 ★\u0000 $star$\u0000 -Sylvester equations with arbitrary coefficients","authors":"Fernando De Terán, Bruno Iannazzo","doi":"10.1112/jlms.70129","DOIUrl":"https://doi.org/10.1112/jlms.70129","url":null,"abstract":"<p>We analyze the consistency and uniqueness of solution of the generalized <span></span><math>\u0000 <semantics>\u0000 <mi>★</mi>\u0000 <annotation>$star$</annotation>\u0000 </semantics></math>-Sylvester equation <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>X</mi>\u0000 <mi>B</mi>\u0000 <mo>+</mo>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>X</mi>\u0000 <mi>★</mi>\u0000 </msup>\u0000 <mi>D</mi>\u0000 <mo>=</mo>\u0000 <mi>E</mi>\u0000 </mrow>\u0000 <annotation>$AXB+CX^star D=E$</annotation>\u0000 </semantics></math>, with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>,</mo>\u0000 <mi>B</mi>\u0000 <mo>,</mo>\u0000 <mi>C</mi>\u0000 <mo>,</mo>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation>$A,B,C, D$</annotation>\u0000 </semantics></math>, and <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> being complex matrices (and <span></span><math>\u0000 <semantics>\u0000 <mi>★</mi>\u0000 <annotation>$star$</annotation>\u0000 </semantics></math> being either the transpose or the conjugate transpose). In particular, we obtain characterizations for the equation to have at most one solution and to be consistent for any right-hand side. Such characterizations are given in terms of spectral properties of the matrix pencils <span></span><math>\u0000 <semantics>\u0000 <mfenced>\u0000 <mtable>\u0000 <mtr>\u0000 <mtd>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <msup>\u0000 <mi>D</mi>\u0000 <mi>★</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mtd>\u0000 <mtd>\u0000 <msup>\u0000 <mi>B</mi>\u0000 <mi>★</mi>\u0000 </msup>\u0000 </mtd>\u0000 </mtr>\u0000 <mtr>\u0000 <mtd>\u0000 <mi>A</mi>\u0000 </mtd>\u0000 <mtd>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 </mtd>\u0000 </mtr>\u0000 </mtable>\u0000 </mfenced>\u0000 <annotation>$left[begin{smallmatrix}lambda D^star & B^star A & lambda Cend{smallmatrix}right]$</annota","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143638898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}