{"title":"Concrete billiard arrays of polynomial type and Leonard systems","authors":"Jimmy Vineyard","doi":"10.1016/j.laa.2025.02.006","DOIUrl":"10.1016/j.laa.2025.02.006","url":null,"abstract":"<div><div>Let <em>d</em> denote a nonnegative integer and let <span><math><mi>F</mi></math></span> denote a field. Let <em>V</em> denote a <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional vector space over <span><math><mi>F</mi></math></span>. Given an ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> of the eigenvalues of a multiplicity-free linear map <span><math><mi>A</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>V</mi></math></span>, we construct a Concrete Billiard Array <span><math><mi>L</mi></math></span> with the property that for <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math></span>, the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mi>th</mi></mrow></msup></math></span> vector on its bottom border is in the <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-eigenspace of <em>A</em>. The Concrete Billiard Array <span><math><mi>L</mi></math></span> is said to have polynomial type. We also show the following. Assume that there exists a Leonard system <span><math><mi>Φ</mi><mo>=</mo><mo>(</mo><mi>A</mi><mo>;</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>;</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>;</mo><msubsup><mrow><mo>{</mo><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></math></span> where <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the primitive idempotent of <em>A</em> corresponding to <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math></span>. Then, we show that after a suitable normalization, the left (resp. right) boundary of <span><math><mi>L</mi></math></span> corresponds to the Φ-split (resp. <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>⇓</mo></mrow></msup></math></span>-split) decomposition of <em>V</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 296-309"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on dual abelian varieties","authors":"Aleksandra Borówka, Paweł Borówka","doi":"10.1016/j.laa.2025.02.008","DOIUrl":"10.1016/j.laa.2025.02.008","url":null,"abstract":"<div><div>For any non-principal polarisation <em>D</em>, we explicitly construct <em>D</em>-polarised abelian variety <em>A</em>, such that its dual abelian variety is not (abstractly) isomorphic to <em>A</em>. For <span><math><mi>dim</mi><mo></mo><mo>(</mo><mi>A</mi><mo>)</mo><mo>></mo><mn>3</mn></math></span> the construction includes examples with submaximal Picard number equal to <span><math><msup><mrow><mo>(</mo><mi>dim</mi><mo></mo><mo>(</mo><mi>A</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span>. As a corollary, we show that a very general non-principally polarised abelian variety is not isomorphic to its dual. Moreover, we show an example of an abelian variety that is isomorphic to its dual, yet it does not admit a principal polarisation.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 458-470"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Piercing intersecting convex sets","authors":"Imre Bárány , Travis Dillon , Dömötör Pálvölgyi , Dániel Varga","doi":"10.1016/j.laa.2025.02.007","DOIUrl":"10.1016/j.laa.2025.02.007","url":null,"abstract":"<div><div>Assume two finite families <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> of convex sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> have the property that <span><math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math></span> for every <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> and <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span>. Is there a constant <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span> (independent of <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span>) such that there is a line intersecting <span><math><mi>γ</mi><mo>|</mo><mi>A</mi><mo>|</mo></math></span> sets in <span><math><mi>A</mi></math></span> or <span><math><mi>γ</mi><mo>|</mo><mi>B</mi><mo>|</mo></math></span> sets in <span><math><mi>B</mi></math></span>? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in <span><math><mi>A</mi></math></span> lie in parallel planes and all sets in <span><math><mi>B</mi></math></span> lie in parallel planes; in fact, one of the two families has a transversal by a single line.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 405-417"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantum state transfer in graphs with tails","authors":"Pierre-Antoine Bernard , Christino Tamon , Luc Vinet , Weichen Xie","doi":"10.1016/j.laa.2025.02.005","DOIUrl":"10.1016/j.laa.2025.02.005","url":null,"abstract":"<div><div>Godsil proved that there is no quantum perfect state transfer (between vertex states) on bounded infinite graphs. We show however there exists quantum perfect state transfer in graphs with tails. The main argument used is a decoupling theorem for eventually-free Jacobi matrices (due to Golinskii). Our results rehabilitate the notion of a dark subspace which had been so far viewed in an unflattering light.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 363-384"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hitesh Kumar, Bojan Mohar , Shivaramakrishna Pragada, Hanmeng Zhan
{"title":"Subdivision and graph eigenvalues","authors":"Hitesh Kumar, Bojan Mohar , Shivaramakrishna Pragada, Hanmeng Zhan","doi":"10.1016/j.laa.2025.01.044","DOIUrl":"10.1016/j.laa.2025.01.044","url":null,"abstract":"<div><div>This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the <em>k</em>-th largest and <em>k</em>-th smallest eigenvalues for any fixed <em>k</em>. Given a graph, we show that after subdividing sufficiently many times, all but <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> eigenvalues of the new graph will lie in the interval <span><math><mo>[</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. We examine the eigenvalues of the new graph outside this interval, and we prove several results that might be of independent interest.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 336-355"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a question of Bourin","authors":"Fuad Kittaneh , Éric Ricard","doi":"10.1016/j.laa.2025.02.004","DOIUrl":"10.1016/j.laa.2025.02.004","url":null,"abstract":"<div><div>We use complex analysis methods to give a partial answer to a question by Bourin in matrix inequalities.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 356-362"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral extremal problems for graphs with bounded clique number","authors":"Tingting Wang, Lihua Feng, Lu Lu","doi":"10.1016/j.laa.2025.01.043","DOIUrl":"10.1016/j.laa.2025.01.043","url":null,"abstract":"<div><div>For a family of graphs <span><math><mi>F</mi></math></span>, a graph is called <span><math><mi>F</mi></math></span>-free if it contains no subgraph isomorphic to any graph in <span><math><mi>F</mi></math></span>. For two integers <em>n</em> and <em>r</em>, let <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> be the set of graphs on <em>n</em> vertices with clique number at most <em>r</em>. Denote by<span><span><span><math><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>G</mi><mspace></mspace><mrow><mtext>is </mtext><mi>F</mi><mtext>-free</mtext></mrow><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the spectral radius of <em>G</em>. Furthermore, denote by <span><math><msub><mrow><mtext>Ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>|</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>G</mi><mtext> is </mtext><mi>F</mi><mtext>-</mtext><mtext>free</mtext><mo>}</mo></math></span> the set of extremal graphs. In this paper, we first give a spectral Erdös-Sós theorem in <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>, that is, for fixed <span><math><mi>k</mi><mo>≥</mo><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and sufficiently large <em>n</em>, if a graph <span><math><mi>G</mi><mo>∈</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> with <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>λ</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>∨</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, then it contains all trees on <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></math></span> vertices or <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn></math></span> vertices unless <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>∨</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 273-295"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On commutators of unipotent matrices of index 2","authors":"Kennett L. Dela Rosa, Juan Paolo C. Santos","doi":"10.1016/j.laa.2025.02.003","DOIUrl":"10.1016/j.laa.2025.02.003","url":null,"abstract":"<div><div>A commutator of unipotent matrices of index 2 is a matrix of the form <span><math><mi>X</mi><mi>Y</mi><msup><mrow><mi>X</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>Y</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, where <em>X</em> and <em>Y</em> are unipotent matrices of index 2, that is, <span><math><mi>X</mi><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>Y</mi><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and <span><math><msup><mrow><mo>(</mo><mi>X</mi><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>Y</mi><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msub><mrow><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msub></math></span>. If <span><math><mi>n</mi><mo>></mo><mn>2</mn></math></span> and <span><math><mi>F</mi></math></span> is a field with <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span>, then it is shown that every <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix over <span><math><mi>F</mi></math></span> with determinant 1 is a product of at most four commutators of unipotent matrices of index 2. Consequently, every <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix over <span><math><mi>F</mi></math></span> with determinant 1 is a product of at most eight unipotent matrices of index 2. Conditions on <span><math><mi>F</mi></math></span> are given that improve the upper bound on the commutator factors from four to three or two. The situation for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> is also considered. This study reveals a connection between factorability into commutators of unipotent matrices and properties of <span><math><mi>F</mi></math></span> such as its characteristic or its set of perfect squares.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 385-404"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nicholas J. Higham , Matthew C. Lettington , Karl Michael Schmidt
{"title":"Symmetry decomposition and matrix multiplication","authors":"Nicholas J. Higham , Matthew C. Lettington , Karl Michael Schmidt","doi":"10.1016/j.laa.2025.01.041","DOIUrl":"10.1016/j.laa.2025.01.041","url":null,"abstract":"<div><div>General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices, the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relate to the pseudoinverses of its component parts. For invertible matrices, this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 310-335"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Additive mappings preserving orthogonality between complex inner product spaces","authors":"Lei Li , Siyu Liu , Antonio M. Peralta","doi":"10.1016/j.laa.2025.01.042","DOIUrl":"10.1016/j.laa.2025.01.042","url":null,"abstract":"<div><div>Let <em>H</em> and <em>K</em> be two complex inner product spaces with dim<span><math><mo>(</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>. We prove that for each non-zero mapping <span><math><mi>A</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>K</mi></math></span> with dense image the following statements are equivalent:<ul><li><span>(<em>a</em>)</span><span><div><em>A</em> is (complex) linear or conjugate-linear mapping and there exists <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span> such that <span><math><mo>‖</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>‖</mo><mo>=</mo><mi>γ</mi><mo>‖</mo><mi>x</mi><mo>‖</mo></math></span>, for all <span><math><mi>x</mi><mo>∈</mo><mi>H</mi></math></span>, that is, <em>A</em> is a positive scalar multiple of a linear or a conjugate-linear isometry;</div></span></li><li><span>(<em>a</em>)</span><span><div>There exists <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> such that one of the next properties holds for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>H</mi></math></span>:<ul><li><span><span><math><mo>(</mo><mi>b</mi><mo>.</mo><mn>1</mn><mo>)</mo></math></span></span><span><div><span><math><mo>〈</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>〉</mo><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>〈</mo><mi>x</mi><mo>|</mo><mi>y</mi><mo>〉</mo></math></span>,</div></span></li><li><span><span><math><mo>(</mo><mi>b</mi><mo>.</mo><mn>1</mn><mo>)</mo></math></span></span><span><div><span><math><mo>〈</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>〉</mo><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>〈</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>〉</mo></math></span>;</div></span></li></ul></div></span></li><li><span>(<em>a</em>)</span><span><div><em>A</em> is linear or conjugate-linear and preserves orthogonality;</div></span></li><li><span>(<em>a</em>)</span><span><div><em>A</em> is additive and preserves orthogonality in both directions;</div></span></li><li><span>(<em>a</em>)</span><span><div><em>A</em> is additive and preserves orthogonality.</div></span></li></ul> This extends to the complex setting a recent generalization of the Koldobsky–Blanco–Turnšek theorem obtained by Wójcik for real normed spaces.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 448-457"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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