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A recurrence relation for the average singular value of a complex Gaussian random matrix. 复高斯随机矩阵的平均奇异值的递推关系。
IF 1 4区 数学
Complex Analysis and Operator Theory Pub Date : 2026-01-01 Epub Date: 2026-08-07 DOI: 10.1007/s11785-026-01988-4
Luís Daniel Abreu
{"title":"A recurrence relation for the average singular value of a complex Gaussian random matrix.","authors":"Luís Daniel Abreu","doi":"10.1007/s11785-026-01988-4","DOIUrl":"10.1007/s11785-026-01988-4","url":null,"abstract":"<p><p>In this brief note, we obtain a recurrence relation for the average singular value <math><mrow><mi>α</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo></mrow> </math> of a complex valued <math><mrow><mi>d</mi> <mo>×</mo> <mi>d</mi></mrow> </math> matrix <math> <mrow><mfrac><mn>1</mn> <msqrt><mi>d</mi></msqrt> </mfrac> <mi>X</mi></mrow> </math> with Gaussian entries, of the form: <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><msub><mi>α</mi> <mi>C</mi></msub> <mfenced><mi>d</mi> <mo>+</mo> <mn>1</mn></mfenced> <mo>=</mo> <msub><mi>α</mi> <mi>C</mi></msub> <mfenced><mi>d</mi></mfenced> <mo>+</mo> <mi>R</mi> <mrow><mo>(</mo> <mi>d</mi> <mo>)</mo></mrow> <mtext>,</mtext></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <i>R</i>(<i>d</i>) can be written as the linear combination of two integrals involving Laguerre polynomials. This reduces to showing that <math><mrow><mi>R</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo> <mo><</mo> <mn>0</mn></mrow> </math> (a non-trivial analysis problem) a 2016 conjecture of Bandeira, Kennedy and Singer, where it is predicted that <math><mrow><mi>α</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo></mrow> </math> decreases monotonically with <i>d</i>, from <math> <mrow><msub><mi>α</mi> <mi>C</mi></msub> <mfenced><mn>1</mn></mfenced> <mo>=</mo> <msqrt><mfrac><mi>π</mi> <mn>4</mn></mfrac> </msqrt> </mrow> </math> , to the <math><mrow><mi>d</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> limit <math><mfrac><mn>8</mn> <mrow><mn>3</mn> <mi>π</mi></mrow> </mfrac> </math> provided by the Marchenko-Pastur distribution. While this may point to a possible path to solving the conjecture, it is still far from a solution, since the two integrals <i>R</i>(<i>d</i>) are non-summable hypergeometric functions, and showing that <math><mrow><mi>R</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo> <mo><</mo> <mn>0</mn></mrow> </math> is a non-trivial analysis problem.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"20 6","pages":"159"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13447555/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148690249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized Conjugate Extension Solutions for Quaternionic Matrix Equations AX*B + CXD = E 四元数矩阵方程AX*B + CXD = E的广义共轭扩展解
4区 数学
Complex Analysis and Operator Theory Pub Date : 2025-09-12 DOI: 10.1007/s11785-025-01797-1
Meng Huo, Lan Zhang
{"title":"Generalized Conjugate Extension Solutions for Quaternionic Matrix Equations AX*B + CXD = E","authors":"Meng Huo, Lan Zhang","doi":"10.1007/s11785-025-01797-1","DOIUrl":"https://doi.org/10.1007/s11785-025-01797-1","url":null,"abstract":"","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"19 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147911085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices. 无界非自伴随Jacobi矩阵的Weyl矩阵透视。
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2025-01-01 Epub Date: 2025-09-23 DOI: 10.1007/s11785-025-01804-5
Benjamin Eichinger, Milivoje Lukić, Giorgio Young
{"title":"A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices.","authors":"Benjamin Eichinger, Milivoje Lukić, Giorgio Young","doi":"10.1007/s11785-025-01804-5","DOIUrl":"https://doi.org/10.1007/s11785-025-01804-5","url":null,"abstract":"<p><p>A new way of encoding a non-self-adjoint Jacobi matrix <i>J</i> by a spectral measure of |<i>J</i>| together with a phase function was described by Pushnitski-Štampach in the bounded case. We present another perspective on this correspondence, based on Weyl functions instead of moments, which simplifies some proofs and generalizes the correspondence to the unbounded case. In particular, we find a bijection between proper Jacobi matrices with positive off-diagonal elements, and a class of spectral data. We prove that this mapping is continuous in a suitable sense. To prove injectivity of the map, we prove a local Borg-Marchenko theorem for unbounded non-self-adjoint Jacobi matrices in this class that may be of independent interest.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"19 7","pages":"194"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12457545/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145151686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Approximation of Dirac Operators with Confining Electrostatic and Lorentz Scalar δ -Shell Potentials. 具有约束静电和洛伦兹标量δ壳势的狄拉克算子的近似。
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2025-01-01 Epub Date: 2025-07-23 DOI: 10.1007/s11785-025-01742-2
Christian Stelzer-Landauer
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Approximation of Dirac Operators with Confining Electrostatic and Lorentz Scalar <ns0:math><ns0:mrow><ns0:mi>δ</ns0:mi></ns0:mrow> </ns0:math> -Shell Potentials.","authors":"Christian Stelzer-Landauer","doi":"10.1007/s11785-025-01742-2","DOIUrl":"10.1007/s11785-025-01742-2","url":null,"abstract":"<p><p>In this paper we continue earlier investigations regarding the approximation of Dirac operators with <math><mi>δ</mi></math> -shell potentials in the norm resolvent sense. In particular, we consider the approximation of Dirac operators with confining electrostatic and Lorentz scalar <math><mi>δ</mi></math> -shell potentials, where the support of the <math><mi>δ</mi></math> -shell potentials is impermeable to particles modelled by such Dirac operators.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"19 6","pages":"139"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12287183/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144735007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Jacobi Operator on $$(-1,1)$$ and Its Various m-Functions $$(-1,1)$$上的雅可比算子及其各种 m 函数
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2024-09-17 DOI: 10.1007/s11785-024-01576-4
Fritz Gesztesy, Lance L. Littlejohn, Mateusz Piorkowski, Jonathan Stanfill
{"title":"The Jacobi Operator on $$(-1,1)$$ and Its Various m-Functions","authors":"Fritz Gesztesy, Lance L. Littlejohn, Mateusz Piorkowski, Jonathan Stanfill","doi":"10.1007/s11785-024-01576-4","DOIUrl":"https://doi.org/10.1007/s11785-024-01576-4","url":null,"abstract":"<p>We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression </p><span>$$begin{aligned} tau _{alpha ,beta } =&amp;- (1-x)^{-alpha } (1+x)^{-beta }(d/dx) big ((1-x)^{alpha +1}(1+x)^{beta +1}big ) (d/dx), &amp;alpha , beta in {mathbb {R}}, , x in (-1,1), end{aligned}$$</span><p>in <span>(L^2big ((-1,1); (1-x)^{alpha } (1+x)^{beta } dxbig ))</span>, <span>(alpha , beta in {mathbb {R}})</span>. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general <span>(eta )</span>-periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced <i>m</i>-functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"34 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Powers of Regular Linear Relations 常规线性关系的幂
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2024-09-14 DOI: 10.1007/s11785-024-01593-3
Houcem Daoud
{"title":"The Powers of Regular Linear Relations","authors":"Houcem Daoud","doi":"10.1007/s11785-024-01593-3","DOIUrl":"https://doi.org/10.1007/s11785-024-01593-3","url":null,"abstract":"<p>In this paper, we analyse properties like nullity, defect, ascent and descent of the powers of regular linear relations. We improve some results related to closure and regularity of powers of linear relations in normed spaces. Further, the obtained results are applied to investigate the descent and essential descent spectrum and to give some stability results.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Entire Symmetric Operators in de Branges–Pontryagin Spaces and a Truncated Matrix Moment Problem 德布朗热-庞特里亚金空间中的全对称算子和截断矩阵矩问题
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2024-09-11 DOI: 10.1007/s11785-024-01591-5
Volodymyr Derkach, Harry Dym
{"title":"Entire Symmetric Operators in de Branges–Pontryagin Spaces and a Truncated Matrix Moment Problem","authors":"Volodymyr Derkach, Harry Dym","doi":"10.1007/s11785-024-01591-5","DOIUrl":"https://doi.org/10.1007/s11785-024-01591-5","url":null,"abstract":"<p>The role of de Branges–Pontryagin spaces as functional models for entire symmetric operators with finite equal deficiency indices and proper gauges in Pontryagin spaces is reviewed and then extended to symmetric operators that are not entire. These results are used to derive an operator representation for generalized Carathéodory functions. Enroute, boundary mappings and the characteristic function of <i>S</i> are defined. Generalized resolvents of symmetric operators <i>S</i> with non dense domains corresponding to single-valued representing extensions <span>({{widetilde{S}} })</span> are characterized in terms of the characteristic function of <i>S</i>. These results are applied to obtain a description of the set of solutions of an indefinite truncated matrix moment problem.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Orthogonal Polynomials Related to Arithmetic and Harmonic Sequences 论与算术序列和谐波序列有关的正交多项式
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2024-09-09 DOI: 10.1007/s11785-024-01589-z
Adhemar Bultheel, Andreas Lasarow
{"title":"On Orthogonal Polynomials Related to Arithmetic and Harmonic Sequences","authors":"Adhemar Bultheel, Andreas Lasarow","doi":"10.1007/s11785-024-01589-z","DOIUrl":"https://doi.org/10.1007/s11785-024-01589-z","url":null,"abstract":"<p>In this paper we study special systems of orthogonal polynomials on the unit circle. More precisely, with a view to the recurrence relations fulfilled by these orthogonal systems, we analyze a link of non-negative arithmetic to harmonic sequences as a main subject. Here, arithmetic sequences appear as coefficients of orthogonal polynomials and harmonic sequences as corresponding Szegő parameters.\u0000</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"274 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Jordan Curve Theorem on a 3D Ball Through Brownian Motion 三维球通过布朗运动的乔丹曲线定理
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2024-09-06 DOI: 10.1007/s11785-024-01595-1
Arni S. R. Srinivasa Rao, Steven G. Krantz
{"title":"A Jordan Curve Theorem on a 3D Ball Through Brownian Motion","authors":"Arni S. R. Srinivasa Rao, Steven G. Krantz","doi":"10.1007/s11785-024-01595-1","DOIUrl":"https://doi.org/10.1007/s11785-024-01595-1","url":null,"abstract":"<p>The Jordan curve theorem states that any simple closed curve in 3<i>D</i> space divides the space into two regions, an interior and an exterior. In this article, we prove the Jordan curve theorem on the boundary of a 3<i>D</i> ball that is inserted in a complex plane bundle. To do so, we make use of the Brownian motion principle, which is a continuous-time and continuous-state stochastic process. We begin by selecting a random point on an arbitrarily chosen complex plane within a bundle <i>G</i> and on the boundary of the 3D ball considered. Using the two-step random process developed on complex planes earlier by Srinivasa Rao (Multilevel contours on bundles of complex planes, 2022), we draw a contour from the initial point to the next point on this plane. We then continue this process until we finish the Jordan curve that connects points on the boundary of a ball within <i>G</i>.\u0000</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Essential Adjointness of Pseudo-Differential Operators on $$mathbb {Z}^n$$ 伪微分算子在 $$mathbb {Z}^n$$ 上的本质相邻性
IF 0.8 4区 数学
Complex Analysis and Operator Theory Pub Date : 2024-09-05 DOI: 10.1007/s11785-024-01597-z
Ognjen Milatovic
{"title":"The Essential Adjointness of Pseudo-Differential Operators on $$mathbb {Z}^n$$","authors":"Ognjen Milatovic","doi":"10.1007/s11785-024-01597-z","DOIUrl":"https://doi.org/10.1007/s11785-024-01597-z","url":null,"abstract":"<p>In the setting of the lattice <span>(mathbb {Z}^n)</span> we consider a pseudo-differential operator <i>A</i> whose symbol belongs to a class defined on <span>(mathbb {Z}^ntimes mathbb {T}^n)</span>, where <span>(mathbb {T}^n)</span> is the <i>n</i>-torus. We realize <i>A</i> as an operator acting between the discrete Sobolev spaces <span>(H^{s_j}(mathbb {Z}^n))</span>, <span>(s_jin mathbb {R})</span>, <span>(j=1,2)</span>, with the discrete Schwartz space serving as the domain of <i>A</i>. We provide a sufficient condition for the essential adjointness of the pair <span>((A,,A^{dagger }))</span>, where <span>(A^{dagger })</span> is the formal adjoint of <i>A</i>.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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