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Tropical Reproducing Kernels and Optimization 热带复制内核与优化
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-05-29 DOI: 10.1007/s00020-024-02769-4
Pierre-Cyril Aubin-Frankowski, Stéphane Gaubert
{"title":"Tropical Reproducing Kernels and Optimization","authors":"Pierre-Cyril Aubin-Frankowski, Stéphane Gaubert","doi":"10.1007/s00020-024-02769-4","DOIUrl":"https://doi.org/10.1007/s00020-024-02769-4","url":null,"abstract":"<p>Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel–Moreau conjugations. This tropical analogue of Aronszajn’s theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a reproducing property. They furthermore include all the Hilbertian kernels classically studied as well as Monge arrays. However, two relevant notions of tropical reproducing kernels must be distinguished, based either on linear or sesquilinear interpretations. The sesquilinear interpretation is the most expressive one, since reproducing spaces then encompass classical max-plus spaces, such as those of (semi)convex functions. In contrast, in the linear interpretation, the reproducing kernels are characterized by a restrictive condition, von Neumann regularity. Finally, we provide a tropical analogue of the “representer theorems”, showing that a class of infinite dimensional regression and interpolation problems admit solutions lying in finite dimensional spaces. We illustrate this theorem by an application to optimal control, in which tropical kernels allow one to represent the value function.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141167439","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}
引用次数: 0
Local Spectral Multiplicity of Selfadjoint Couplings with General Interface Conditions 具有一般界面条件的自相关耦合的局部谱多重性
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-05-22 DOI: 10.1007/s00020-024-02767-6
Sergey Simonov, Harald Woracek
{"title":"Local Spectral Multiplicity of Selfadjoint Couplings with General Interface Conditions","authors":"Sergey Simonov, Harald Woracek","doi":"10.1007/s00020-024-02767-6","DOIUrl":"https://doi.org/10.1007/s00020-024-02767-6","url":null,"abstract":"<p>We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schrödinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of “selfadjoint” interface conditions, subject to an assumption which is generically satisfied, is considered. We determine the spectral multiplicity function on the singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148960","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}
引用次数: 0
On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach 论紧凑算子的换元：安德森方法的概括与局限
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-05-17 DOI: 10.1007/s00020-024-02764-9
Jireh Loreaux, Sasmita Patnaik, Srdjan Petrovic, Gary Weiss
{"title":"On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach","authors":"Jireh Loreaux, Sasmita Patnaik, Srdjan Petrovic, Gary Weiss","doi":"10.1007/s00020-024-02764-9","DOIUrl":"https://doi.org/10.1007/s00020-024-02764-9","url":null,"abstract":"<p>We offer a new perspective and some advances on the 1971 Pearcy-Topping problem: is every compact operator a commutator of compact operators? Our goal is to analyze and generalize the 1970’s work in this area of Joel Anderson. We reduce the general problem to a simpler sequence of finite matrix equations with norm constraints, while at the same time developing strategies for counterexamples. Our approach is to ask which compact operators are commutators AB − BA of compact operators A,B; and to analyze the implications of Joel Anderson’s contributions to this problem. By extending the techniques of Anderson, we obtain new classes of operators that are commutators of compact operators beyond those obtained by the second and the fourth author. We also found obstructions to extending Anderson’s techniques to obtain any positive compact operator as a commutator of compact operators. Some of these constraints involve general block-tridiagonal matrix forms for operators and some involve B(H)-ideal constraints. Finally, we provide some necessary conditions for the Pearcy-Topping problem involving singular numbers and B(H)-ideal constraints.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062504","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}
引用次数: 0
Subhomogeneous Operator Systems and Classification of Operator Systems Generated by $$Lambda $$ -Commuting Unitaries 亚均质算子系统和由 $$Lambda $$ - 通约单元生成的算子系统分类
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-05-05 DOI: 10.1007/s00020-024-02765-8
Ran Kiri
{"title":"Subhomogeneous Operator Systems and Classification of Operator Systems Generated by $$Lambda $$ -Commuting Unitaries","authors":"Ran Kiri","doi":"10.1007/s00020-024-02765-8","DOIUrl":"https://doi.org/10.1007/s00020-024-02765-8","url":null,"abstract":"<p>A unital <span>(C^*)</span>-algebra is called <i>N</i>-subhomogeneous if its irreducible representations are finite dimensional with dimension at most <i>N</i>. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering <span>(text {UCP }(mathcal {S}))</span> which is the matrix state space associated with an operator system <span>(mathcal {S})</span> and identifying the boundary representations as absolute matrix extreme points. We show that two <i>N</i>-subhomogeneous operator systems are completely order equivalent if and only if they are <i>N</i>-order equivalent. Moreover, we show that a unital <i>N</i>-positive map into a finite dimensional <i>N</i>-subhomogeneous operator system is completely positive. We apply these tools to classify pairs of <i>q</i>-commuting unitaries up to <span>(*)</span>-isomorphism. Similar results are obtained for operator systems related to higher dimensional non-commutative tori.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887891","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}
引用次数: 0
Real Structure in Operator Spaces, Injective Envelopes and G-spaces 算子空间、注入包络和 G 空间中的实结构
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-04-24 DOI: 10.1007/s00020-024-02766-7
David P. Blecher, Arianna Cecco, Mehrdad Kalantar
{"title":"Real Structure in Operator Spaces, Injective Envelopes and G-spaces","authors":"David P. Blecher, Arianna Cecco, Mehrdad Kalantar","doi":"10.1007/s00020-024-02766-7","DOIUrl":"https://doi.org/10.1007/s00020-024-02766-7","url":null,"abstract":"<p>We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and <span>(C^*)</span>-envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.\u0000</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"42 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806682","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}
引用次数: 0
Sufficient Criteria for Stabilization Properties in Banach Spaces 巴拿赫空间稳定特性的充分标准
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-04-08 DOI: 10.1007/s00020-024-02762-x
Michela Egidi, Dennis Gallaun, Christian Seifert, Martin Tautenhahn
{"title":"Sufficient Criteria for Stabilization Properties in Banach Spaces","authors":"Michela Egidi, Dennis Gallaun, Christian Seifert, Martin Tautenhahn","doi":"10.1007/s00020-024-02762-x","DOIUrl":"https://doi.org/10.1007/s00020-024-02762-x","url":null,"abstract":"<p>We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in <span>(L_p(mathbb {R}^d))</span> for <span>(pin [1,infty ))</span> and thick control sets.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"31 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562405","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}
引用次数: 0
Ideal Structure of Nica-Toeplitz Algebras Nica-Toeplitz 算法的理想结构
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-03-27 DOI: 10.1007/s00020-024-02761-y
Boris Bilich
{"title":"Ideal Structure of Nica-Toeplitz Algebras","authors":"Boris Bilich","doi":"10.1007/s00020-024-02761-y","DOIUrl":"https://doi.org/10.1007/s00020-024-02761-y","url":null,"abstract":"<p>We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra <span>(mathcal {N}mathcal {T}(X))</span> of a product system (<i>A</i>, <i>X</i>) over <span>(mathbb {N}^n)</span>. We obtain a clear description of <i>X</i>-invariant ideals in <i>A</i>, that is, restrictions of gauge-invariant ideals in <span>(mathcal {Nhspace{-1.111pt}T}(X))</span> to <i>A</i>. The main result is a classification of gauge-invariant ideals in <span>(mathcal {Nhspace{-1.111pt}T}(X))</span> for a proper product system in terms of families of ideals in <i>A</i>. We also apply our results to higher-rank graphs.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"139 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314909","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}
引用次数: 0
Recovering the Shape of an Equilateral Quantum Tree by Two Spectra 通过两个光谱恢复等边量子树的形状
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-03-27 DOI: 10.1007/s00020-024-02759-6
Vyacheslav Pivovarchik
{"title":"Recovering the Shape of an Equilateral Quantum Tree by Two Spectra","authors":"Vyacheslav Pivovarchik","doi":"10.1007/s00020-024-02759-6","DOIUrl":"https://doi.org/10.1007/s00020-024-02759-6","url":null,"abstract":"<p>We show how to find the shape of an equilateral tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm–Liouville equation. In case of snowflake trees the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"2010 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314929","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}
引用次数: 0
In Memoriam Heinz Langer (08.08.1935-25.01.2024) 纪念海因茨-朗格（1935 年 8 月 8 日-2024 年 1 月 25 日）
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-03-23 DOI: 10.1007/s00020-024-02763-w
C. Tretter
{"title":"In Memoriam Heinz Langer (08.08.1935-25.01.2024)","authors":"C. Tretter","doi":"10.1007/s00020-024-02763-w","DOIUrl":"https://doi.org/10.1007/s00020-024-02763-w","url":null,"abstract":"","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"121 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198327","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}
引用次数: 0
On Operator Valued Haar Unitaries and Bipolar Decompositions of R-diagonal Elements 论R对角元素的算子值哈尔单元和双极分解
IF 0.8 3区 数学
Integral Equations and Operator Theory Pub Date : 2024-02-26 DOI: 10.1007/s00020-024-02760-z
{"title":"On Operator Valued Haar Unitaries and Bipolar Decompositions of R-diagonal Elements","authors":"","doi":"10.1007/s00020-024-02760-z","DOIUrl":"https://doi.org/10.1007/s00020-024-02760-z","url":null,"abstract":"<h3>Abstract</h3> <p>In the context of operator valued W<span> <span>(^*)</span> </span>-free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as <em>vx</em> where <em>x</em> is self-adjoint and <em>v</em> is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that <em>v</em> and <em>x</em> are <span> <span>(*)</span> </span>-free from each other. In particular, we prove, when <span> <span>(B={textbf{C}}^2)</span> </span>, that if a <em>B</em>-valued circular element has a free bipolar decomposition with <em>v</em> unitary, then it has one where <em>v</em> normalizes <em>B</em>.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"2015 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969476","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}
引用次数: 0
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