Linear Algebra and its Applications最新文献

{"title":"Diagonalization of Operator functions by algebraic methods","authors":"Matthias Stiefenhofer","doi":"10.1016/j.laa.2025.07.013","DOIUrl":"10.1016/j.laa.2025.07.013","url":null,"abstract":"<div><div>We give conditions for local diagonalization of an analytic operator family <span><math><mi>L</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> according to <span><math><mi>L</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow><mo>⋅</mo><mi>Δ</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow><mo>⋅</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> with diagonal operator polynomial <span><math><mi>Δ</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> and analytic near identity bijections <span><math><mi>ψ</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> and <span><math><mi>ϕ</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span>. The family <span><math><mi>L</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> is acting between real or complex Banach spaces <em>B</em> and <span><math><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span>.</div><div>The basic assumption is given by stabilization of the Jordan chains at length <em>k</em> in the sense that no root elements with finite rank above <em>k</em> are allowed to exist. Jordan chains with infinite rank may appear. Decompositions of the linear spaces <em>B</em> and <span><math><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> are constructed with corresponding subspaces assumed to be closed. These assumptions ensure finite pole order equal to <em>k</em> of the generalized inverse <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> at <span><math><mi>ε</mi><mo>=</mo><mn>0</mn></math></span>. The Smith form and smooth continuation of kernels and ranges of <span><math><mi>L</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> to appropriate limit spaces at <span><math><mi>ε</mi><mo>=</mo><mn>0</mn></math></span> arise immediately.</div><div>An algebraically oriented and self-contained approach is used, based on a recursion that allows for construction of power series solutions of <span><math><mi>L</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow><mo>⋅</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></span>. The power series solutions are convergent, as soon as analyticity of <span><math><mi>L</mi><mrow><mo>(</mo><mi>ε</mi><mo>)</mo></mrow></math></span> and continuity of related projections are assumed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 319-354"},"PeriodicalIF":1.0,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144670587","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 Erdős matrices and Marcus–Ree inequality","authors":"Aman Kushwaha ,&nbsp;Raghavendra Tripathi","doi":"10.1016/j.laa.2025.07.012","DOIUrl":"10.1016/j.laa.2025.07.012","url":null,"abstract":"<div><div>In 1959, Marcus and Ree proved that any bistochastic matrix <em>A</em> satisfies<span><span><span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>≔</mo><munder><mi>max</mi><mrow><mi>σ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></munder><mo>⁡</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>A</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>σ</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>A</mi><msup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>≥</mo><mn>0</mn><mspace></mspace><mo>.</mo></math></span></span></span> Erdős asked to characterize the bistochastic matrices satisfying <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. This problem remains largely open, and very recently, a complete list of such matrices was obtained in dimension <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> by Bouthat, Mashreghi, and Morneau-Guérin. Soon after, Tripathi proved that there were only finitely many such matrices in any dimension <em>n</em>. In this paper, we continue the investigation initiated in these two works. We characterize all <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> bistochastic matrices satisfying <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. Furthermore, we show that for <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>α</mi></math></span> has uncountably many solutions when <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>4</mn><mo>)</mo></math></span>. This answers a question raised in (Tripathi, 2025 <span><span>[16]</span></span>). We also extend the Marcus–Ree inequality to infinite bistochastic arrays and bistochastic kernels. Our investigation into <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> Erdős matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 223-247"},"PeriodicalIF":1.0,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657316","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":"Leonard pairs arising from dual polar spaces","authors":"Bo Hou ,&nbsp;Runxian Zhang ,&nbsp;Lihang Hou","doi":"10.1016/j.laa.2025.07.010","DOIUrl":"10.1016/j.laa.2025.07.010","url":null,"abstract":"<div><div>A Leonard pair is an ordered pair of diagonalizable linear transformations on a finite-dimensional vector space such that each of the two transformations acts on an eigenbasis for the other one in an irreducible tridiagonal form. In this paper, we consider Leonard pairs arising from a class of graded posets called dual polar spaces. Let <em>P</em> be a dual polar space with rank <em>N</em> and let <em>T</em> be its incident algebra (over the complex field) generated by the raising matrix <em>R</em>, the lowering matrix <em>L</em> and the projection matrices <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mo>(</mo><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>N</mi><mo>)</mo></math></span>. Define two elements <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> of <em>T</em>: <span><math><mi>A</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>R</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msubsup><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mi>L</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msubsup><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Assume <span><math><mi>N</mi><mo>≥</mo><mn>8</mn></math></span>. We first give a necessary and sufficient condition for <em>A</em> and <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> to satisfy the so-called tridiagonal relations. Then these results allow us to further display a necessary and sufficient condition for <em>A</em> and <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acting on any irreducible <em>T</em>-module as a Leonard pair.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 115-134"},"PeriodicalIF":1.0,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633844","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":"Real skew-symmetric and periodic skew-symmetric tridiagonal matrices with prescribed spectral data","authors":"Yu Zeng ,&nbsp;Wei-Ru Xu ,&nbsp;Natália Bebiano","doi":"10.1016/j.laa.2025.07.009","DOIUrl":"10.1016/j.laa.2025.07.009","url":null,"abstract":"<div><div>Skew-symmetric matrices have various applications in physics, one of which is in the gyroscopic system. In this paper, we retrieve a unique real skew-symmetric tridiagonal matrix from the maximal and minimal imaginary parts of the eigenvalues of all its leading principal submatrices. For <em>n</em> even, we then reconstruct an <em>n</em>-by-<em>n</em> real periodic skew-symmetric tridiagonal matrix from the imaginary parts of all its eigenvalues, and those of the leading principal submatrix of order <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, and a given positive number. The necessary and sufficient conditions for the existence of such matrices are given, and we show that the total number of possible <em>n</em>-by-<em>n</em> periodic skew-symmetric tridiagonal matrices is at most <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></msup></math></span>, where <em>s</em> is the number of common elements in the two prescribed spectra. The proofs of the obtained results provide algorithmic procedures for the aimed reconstructions, which are supported by some illustrative numerical experiments.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 198-222"},"PeriodicalIF":1.0,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633762","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":"Matrix invertible extensions over commutative rings. Part II: Determinant liftability","authors":"Grigore Călugăreanu ,&nbsp;Horia F. Pop ,&nbsp;Adrian Vasiu","doi":"10.1016/j.laa.2025.07.008","DOIUrl":"10.1016/j.laa.2025.07.008","url":null,"abstract":"<div><div>A unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix <em>A</em> with entries in a commutative ring <em>R</em> is called weakly determinant liftable if there exists a matrix <em>B</em> congruent to <em>A</em> modulo <span><math><mi>R</mi><mi>det</mi><mo>⁡</mo><mo>(</mo><mi>A</mi><mo>)</mo></math></span> and <span><math><mi>det</mi><mo>⁡</mo><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>; if we can choose <em>B</em> to be unimodular, then <em>A</em> is called determinant liftable. If <em>A</em> is extendable to an invertible <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, then <em>A</em> is weakly determinant liftable. If <em>A</em> is simply extendable (i.e., we can choose <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> such that its <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> entry is 0), then <em>A</em> is determinant liftable. We present necessary and/or sufficient criteria for <em>A</em> to be (weakly) determinant liftable and we use them to show that if <em>R</em> is a <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ring in the sense of Part I (resp. is a pre-Schreier domain), then <em>A</em> is simply extendable (resp. extendable) iff it is determinant liftable (resp. weakly determinant liftable). As an application we show that each <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span> domain (as defined by Lorenzini) is an elementary divisor domain.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 172-197"},"PeriodicalIF":1.0,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633761","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 logical implication between two conjectures on matrix permanents","authors":"Léo Pioge ,&nbsp;Kamil K. Pietrasz ,&nbsp;Benoit Seron ,&nbsp;Leonardo Novo ,&nbsp;Nicolas J. Cerf","doi":"10.1016/j.laa.2025.07.011","DOIUrl":"10.1016/j.laa.2025.07.011","url":null,"abstract":"<div><div>We prove a logical implication between two old conjectures stated by Bapat and Sunder about the permanent of positive semidefinite matrices. Although Drury has recently disproved both conjectures, this logical implication yields a nontrivial link between two seemingly unrelated conditions that a positive semidefinite matrix may fulfill. As a corollary, the classes of matrices that are known to obey the first conjecture are then immediately proven to obey the second one. Conversely, we uncover new counterexamples to the first conjecture by exhibiting a previously unknown type of counterexamples to the second conjecture. Interestingly, such a relationship between these two mathematical conjectures appears from considerations on their quantum physics implications.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 309-318"},"PeriodicalIF":1.0,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713803","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":"The dynamical Cuntz semigroup and ideal-free quotients of Cuntz semigroups","authors":"Joan Bosa ,&nbsp;Francesc Perera ,&nbsp;Jianchao Wu ,&nbsp;Joachim Zacharias","doi":"10.1016/j.laa.2025.07.006","DOIUrl":"10.1016/j.laa.2025.07.006","url":null,"abstract":"<div><div>We develop a theory of general quotients for W- and Cu-semigroups beyond the case of quotients by ideals. To this end, we introduce the notion of a normal pair, which allows us to take quotients of W-semigroups in a similar way as normal subgroups arise as kernels of group homomorphisms.</div><div>We use this to define the dynamical Cuntz semigroup as the universal object induced from an action of a group <em>G</em> on a W-semigroup. In the C*-algebraic framework, under mild assumptions, the universality of this dynamical invariant helps us tap into the structure of the Cuntz semigroup of crossed product C*-algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 248-308"},"PeriodicalIF":1.0,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657317","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":"Extending Elman's bound for GMRES","authors":"Mark Embree","doi":"10.1016/j.laa.2025.07.007","DOIUrl":"10.1016/j.laa.2025.07.007","url":null,"abstract":"<div><div>If the numerical range of a matrix is contained in the open right half of the complex plane, the GMRES algorithm for solving linear systems will reduce the norm of the residual at every iteration. In his Ph.D. dissertation, Howard Elman derived a bound that guarantees convergence. When the numerical range contains the origin, GMRES need not make progress at every step and Elman's bound does not apply, even if all the eigenvalues have positive real part. By solving a Lyapunov equation, one can construct an inner product in which the numerical range is contained in the open right half-plane. One can then bound GMRES (run in the standard Euclidean norm) by applying Elman's bound (or most other GMRES bounds) in this new inner product, at the cost of a multiplicative constant that characterizes the distortion caused by the change of inner product. Using Lyapunov inverse iteration, one can build a family of such inner products, trading off the location of the numerical range with the size of constant. This approach complements techniques that Greenbaum and colleagues have recently proposed for excising the origin from the numerical range to gain greater insight into the convergence of GMRES for nonnormal matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"726 ","pages":"Pages 54-70"},"PeriodicalIF":1.0,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702566","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 generalization of the Böttcher-Wenzel inequality for three rectangular matrices","authors":"Motoyuki Nobori","doi":"10.1016/j.laa.2025.07.005","DOIUrl":"10.1016/j.laa.2025.07.005","url":null,"abstract":"<div><div>Let <span><math><mi>m</mi><mo>,</mo><mi>n</mi></math></span> be positive integers. For all <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> complex matrices <span><math><mi>A</mi><mo>,</mo><mi>C</mi></math></span> and an <span><math><mi>n</mi><mo>×</mo><mi>m</mi></math></span> matrix <em>B</em>, we define a generalized commutator as <span><math><mi>A</mi><mi>B</mi><mi>C</mi><mo>−</mo><mi>C</mi><mi>B</mi><mi>A</mi></math></span>. We estimate the Frobenius norm of it, and finally get the inequality, which is a generalization of the Böttcher-Wenzel inequality. If <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> or <span><math><mi>m</mi><mo>=</mo><mn>1</mn></math></span>, then the Frobenius norm of <span><math><mi>A</mi><mi>B</mi><mi>C</mi><mo>−</mo><mi>C</mi><mi>B</mi><mi>A</mi></math></span> can be estimated with a tighter upper bound.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 135-144"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633845","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":"Varieties of group-graded algebras of proper central exponent greater than two","authors":"Francesca S. Benanti ,&nbsp;Angela Valenti","doi":"10.1016/j.laa.2025.07.001","DOIUrl":"10.1016/j.laa.2025.07.001","url":null,"abstract":"<div><div>Let <em>F</em> be a field of characteristic zero and let <span><math><mi>V</mi></math></span> be a variety of associative <em>F</em>-algebras graded by a finite abelian group <em>G</em>. To a variety <span><math><mi>V</mi></math></span> is associated a numerical sequence called the sequence of proper central <em>G</em>-codimensions, <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>δ</mi></mrow></msubsup><mo>(</mo><mi>V</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>. Here <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>δ</mi></mrow></msubsup><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is the dimension of the space of multilinear proper central <em>G</em>-polynomials in <em>n</em> fixed variables of any algebra <em>A</em> generating the variety <span><math><mi>V</mi></math></span>. Such sequence gives information on the growth of the proper central <em>G</em>-polynomials of <em>A</em> and in <span><span>[21]</span></span> it was proved that <span><math><mi>e</mi><mi>x</mi><msup><mrow><mi>p</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>δ</mi></mrow></msup><mo>(</mo><mi>V</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><mroot><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>δ</mi></mrow></msubsup><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mroot></math></span> exists and is an integer called the proper central <em>G</em>-exponent.</div><div>The aim of this paper is to characterize the varieties of associative <em>G</em>-graded algebras of proper central <em>G</em>-exponent greater than two. To this end we construct a finite list of <em>G</em>-graded algebras and we prove that <span><math><mi>e</mi><mi>x</mi><msup><mrow><mi>p</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>δ</mi></mrow></msup><mo>(</mo><mi>V</mi><mo>)</mo><mo>&gt;</mo><mn>2</mn></math></span> if and only if at least one of the algebras belongs to <span><math><mi>V</mi></math></span>.</div><div>Matching this result with the characterization of the varieties of almost polynomial growth given in <span><span>[11]</span></span>, we obtain a characterization of the varieties of proper central <em>G</em>-exponent equal to two.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"725 ","pages":"Pages 145-171"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633846","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}