{"title":"Application of operator theory for the collatz conjecture","authors":"Takehiko Mori","doi":"10.1007/s43036-025-00425-1","DOIUrl":"10.1007/s43036-025-00425-1","url":null,"abstract":"<div><p>The Collatz map (or the <span>(3n{+}1)</span>-map) <i>f</i> is defined on positive integers by setting <i>f</i>(<i>n</i>) equal to <span>(3n+1)</span> when <i>n</i> is odd and <i>n</i>/2 when <i>n</i> is even. The Collatz conjecture states that starting from any positive integer <i>n</i>, some iterate of <i>f</i> takes value 1. In this study, we discuss formulations of the Collatz conjecture by <span>(C^{*})</span>-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the <span>(C^{*})</span>-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated <span>(C^{*})</span>-algebras.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Comparing the ill-posedness for linear operators in Hilbert spaces","authors":"Peter Mathé, Bernd Hofmann","doi":"10.1007/s43036-025-00422-4","DOIUrl":"10.1007/s43036-025-00422-4","url":null,"abstract":"<div><p>The difficulty for solving ill-posed linear operator equations in Hilbert space is reflected by the strength of ill-posedness of the governing operator, and the inherent solution smoothness. In this study we focus on the ill-posedness of the operator, and we propose a partial ordering for the class of all bounded linear operators which lead to ill-posed operator equations. For compact linear operators, there is a simple characterization in terms of the decay rates of the singular values. In the context of the validity of the spectral theorem the partial ordering can also be understood. We highlight that range inclusions yield partial ordering, and we discuss cases when compositions of compact and non-compact operators occur. Several examples complement the theoretical results.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143430995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Using the Baire category theorem to explore Lions problem for quasi-Banach spaces","authors":"A. G. Aksoy, J. M. Almira","doi":"10.1007/s43036-025-00423-3","DOIUrl":"10.1007/s43036-025-00423-3","url":null,"abstract":"<div><p>Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire category theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple <span>((A_0, A_1).)</span> Lions problem, posed in 1960s, is to prove that different parameters <span>((theta ,p))</span> produce different interpolation spaces <span>((A_0, A_1)_{theta , p}.)</span> We first establish conditions on <span>(A_0)</span> and <span>(A_1)</span> so that interpolation spaces of this couple are strictly intermediate spaces between <span>(A_0+A_1)</span> and <span>(A_0cap A_1.)</span> This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if <span>(pne p^*)</span> the operator ideals <span>({mathcal {L}}^{(a)}_{p,q}(X,Y),)</span> <span>({mathcal {L}}^{(a)}_{p^*,q^*}(X,Y))</span> generated by approximation numbers are distinct. Moreover, for any fixed <i>p</i>, either all operator ideals <span>({mathcal {L}}^{(a)}_{p,q}(X,Y))</span> collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch’s problem for operator ideals based on general <i>s</i>-numbers. However, the BCT can be used to prove a lethargy result for arbitrary <i>s</i>-numbers which guarantees that, under very minimal conditions on <i>X</i>, <i>Y</i>, the space <span>({mathcal {L}}^{(s)}_{p,q}(X,Y))</span> is strictly embedded into <span>({mathcal {L}}^{mathcal {A}}(X,Y).)</span></p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-025-00423-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143423476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tanise Carnieri Pierin, Ruth Nascimento Ferreira, Fernando Borges, Bruno Leonardo Macedo Ferreira
{"title":"About the additivity of a nonlinear mixed (*)-Jordan type derivation defined on an alternative (*)-algebra","authors":"Tanise Carnieri Pierin, Ruth Nascimento Ferreira, Fernando Borges, Bruno Leonardo Macedo Ferreira","doi":"10.1007/s43036-025-00424-2","DOIUrl":"10.1007/s43036-025-00424-2","url":null,"abstract":"<div><p>For an alternative <span>(*)</span>-algebra <i>A</i> under some additional hypothesis, we prove that a map from <i>A</i> into itself is a nonlinear mixed <span>(*)</span>-Jordan type derivation if and only is an additive <span>(*)</span>-derivation. As consequence, some results on the complex octonion algebra, associative <span>(*)</span>-algebras, and <span>(W^*)</span>-factor algebras were obtained.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143423475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Commuting families of polygonal type operators on Hilbert space","authors":"Christian Le Merdy, M. N. Reshmi","doi":"10.1007/s43036-024-00407-9","DOIUrl":"10.1007/s43036-024-00407-9","url":null,"abstract":"<div><p>Let <span>(T:Hrightarrow H)</span> be a bounded operator on Hilbert space <i>H</i>. We say that <i>T</i> has a polygonal type if there exists an open convex polygon <span>(Delta subset {mathbb {D}})</span>, with <span>(overline{Delta }cap {mathbb {T}}ne emptyset )</span>, such that the spectrum <span>(sigma (T))</span> is included in <span>(overline{Delta })</span> and the resolvent <i>R</i>(<i>z</i>, <i>T</i>) satisfies an estimate <span>(Vert R(z,T)Vert lesssim max {vert z-xi vert ^{-1},:, xi in overline{Delta }cap {mathbb {T}}})</span> for <span>(zin overline{mathbb {D}}^c)</span>. The class of polygonal type operators (which goes back to De Laubenfels and Franks–McIntosh) contains the class of Ritt operators. Let <span>(T_1,ldots ,T_d)</span> be commuting operators on <i>H</i>, with <span>(dge 3)</span>. We prove functional calculus properties of the <i>d</i>-tuple <span>((T_1,ldots ,T_d))</span> under various assumptions involving poygonal type. The main ones are the following. (1) If the operator <span>(T_k)</span> is a contraction for all <span>(k=1,ldots ,d)</span> and if <span>(T_1,ldots ,T_{d-2})</span> have a polygonal type, then <span>((T_1,ldots ,T_d))</span> satisfies a generalized von Neumann inequality <span>(Vert phi (T_1,ldots ,T_d)Vert le CVert phi Vert _{infty ,{mathbb {D}}^d})</span> for polynomials <span>(phi )</span> in <i>d</i> variables; (2) If <span>(T_k)</span> is polynomially bounded with a polygonal type for all <span>(k=1,ldots ,d)</span>, then there exists an invertible operator <span>(S:Hrightarrow H)</span> such that <span>(Vert S^{-1}T_kSVert le 1)</span> for all <span>(k=1,ldots ,d)</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143361809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Aspects of equivariant KK-theory in its generators and relations picture","authors":"Bernhard Burgstaller","doi":"10.1007/s43036-024-00412-y","DOIUrl":"10.1007/s43036-024-00412-y","url":null,"abstract":"<div><p>We consider the universal additive category derived from the category of equivariant separable <span>(C^*)</span>-algebras by introducing homotopy invariance, stability and split-exactness. We show that morphisms in that category permit a particular simple form, thus obtaining the universal property of <span>(KK^G)</span>-theory for <i>G</i> a locally compact group, or a locally compact groupoid with compact base space, or a countable inverse semigroup as a byproduct.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143107805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bilinear Fourier multipliers on Orlicz spaces as a dual space","authors":"Serap Öztop, Rüya Üster","doi":"10.1007/s43036-024-00419-5","DOIUrl":"10.1007/s43036-024-00419-5","url":null,"abstract":"<div><p>Let <i>G</i> be a locally compact abelian group with Haar measure and <span>(Phi )</span> be a Young function. In this paper we characterize the space of bilinear Fourier multipliers as a dual space of a certain Banach algebras for Orlicz spaces.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Representation of sequence classes by operator ideals: Part II","authors":"Geraldo Botelho, Ariel S. Santiago","doi":"10.1007/s43036-025-00421-5","DOIUrl":"10.1007/s43036-025-00421-5","url":null,"abstract":"<div><p>In this paper we continue the investigation of classes of vector-valued sequences that are represented by Banach operator ideals. By a procedure we mean a correspondence <span>(X mapsto X^{textrm{new}})</span> that assigns a sequence class <span>(X^{textrm{new}})</span> built upon a given sequence class <i>X</i>. The general question is whether or not <span>(X^{textrm{new}})</span> is ideal-representable whenever <i>X</i> is. We address this question for three already studied procedures, namely, <span>(X mapsto X^{textrm{u}})</span>, <span>(X mapsto X^{textrm{dual}})</span> and <span>(X mapsto X^{textrm{fd}})</span>. Applications of the solutions of these problem will provide new concrete examples of ideal-representable sequence classes.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Charles R. Johnson, António Leal-Duarte, Carlos M. Saiago
{"title":"Fundamental graphs for the maximum multiplicity of an eigenvalue among Hermitian matrices with a given graph","authors":"Charles R. Johnson, António Leal-Duarte, Carlos M. Saiago","doi":"10.1007/s43036-025-00420-6","DOIUrl":"10.1007/s43036-025-00420-6","url":null,"abstract":"<div><p>Our purpose is to identify the graphs that are “fundamental” for the maximum multiplicity problem for Hermitian matrices with a given undirected simple graph. Like paths for trees, these are the special graphs to which the maximum multiplicity problem may be reduced. These are the graphs for which maximum multiplicity implies that all vertices are downers. Examples include cycles and complete graphs, and several more are identified, using the theory developed herein. All the unicyclic graphs that are fundamental, are explicitly identified. We also list those graphs with two edges added to a tree, and their maximum multiplicities, which we have found so far to be fundamental. A formula for maximum multiplicity is given based on fundamental graphs.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-025-00420-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The (C^*)-algebra of the Heisenberg motion groups (U(d) < imes mathbb {H}_d.)","authors":"Hedi Regeiba, Aymen Rahali","doi":"10.1007/s43036-024-00417-7","DOIUrl":"10.1007/s43036-024-00417-7","url":null,"abstract":"<div><p>Let <span>(mathbb {H}_d:=mathbb {C}^dtimes mathbb {R},)</span> <span>((din mathbb {N}^*))</span> be the <span>(2d+1)</span>-dimensional Heisenberg group and we denote by <i>U</i>(<i>d</i>) (the unitary group) the maximal compact connected subgroup of <span>(Aut(mathbb {H}_d),)</span> the group of automorphisms of <span>(mathbb {H}_d.)</span> Let <span>(G_d:=U(d) < imes mathbb {H}_d)</span> be the Heisenberg motion group. In this work, we describe the <span>(C^*)</span>-algebra <span>(C^*(G_d),)</span> of <span>(G_d)</span> in terms of an algebra of operator fields defined over its dual space <span>(widehat{G_d}.)</span> This result generalizes a previous result in Ludwig and Regeiba (Complex Anal Oper Theory 13(8):3943–3978, 2019).</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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