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On the Cesàro hypercyclic linear relations 关于塞萨罗超循环线性关系
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-10 DOI: 10.1007/s43036-024-00387-w
Ali Ech-Chakouri, Hassane Zguitti
{"title":"On the Cesàro hypercyclic linear relations","authors":"Ali Ech-Chakouri,&nbsp;Hassane Zguitti","doi":"10.1007/s43036-024-00387-w","DOIUrl":"10.1007/s43036-024-00387-w","url":null,"abstract":"<div><p>In this paper, we generalize and investigate the concept of Cesàro hypercyclicity of linear operators for linear relations. In addition, we provide new characterizations and properties for this concept.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142411141","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}
引用次数: 0
Spectral theory for fractal pseudodifferential operators 分形伪微分算子的谱理论
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-09 DOI: 10.1007/s43036-024-00381-2
Hans Triebel
{"title":"Spectral theory for fractal pseudodifferential operators","authors":"Hans Triebel","doi":"10.1007/s43036-024-00381-2","DOIUrl":"10.1007/s43036-024-00381-2","url":null,"abstract":"<div><p>The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator <span>(T^mu _tau )</span>, </p><div><div><span>$$begin{aligned} big ( T^mu _tau fbig )(x) = int _{{{mathbb {R}}}^n} e^{-ixxi } , tau (x,xi ) , big ( fmu big )^vee (xi ) , {mathrm d}xi , qquad xin {{mathbb {R}}}^n, end{aligned}$$</span></div></div><p>in suitable special Besov spaces <span>(B^s_p ({{mathbb {R}}}^n) = B^s_{p,p} ({{mathbb {R}}}^n))</span>, <span>(s&gt;0)</span>, <span>(1&lt;p&lt;infty )</span>. Here <span>(tau (x,xi ))</span> are the symbols of (smooth) pseudodifferential operators belonging to appropriate Hörmander classes <span>(Psi ^sigma _{1, delta } ({{mathbb {R}}}^n))</span>, <span>(sigma &lt;0)</span>, <span>(0 le delta le 1)</span> (including the exotic case <span>(delta =1)</span>) whereas <span>(mu )</span> is the Hausdorff measure of a compact <i>d</i>–set <span>(Gamma )</span> in <span>({{mathbb {R}}}^n)</span>, <span>(0&lt;d&lt;n)</span>. This extends previous assertions for the positive-definite selfadjoint fractal differential operator <span>((textrm{id}- Delta )^{sigma /2} mu )</span> based on Hilbert space arguments in the context of suitable Sobolev spaces <span>(H^s ({{mathbb {R}}}^n) = B^s_2 ({{mathbb {R}}}^n))</span>. We collect the outcome in the <b>Main Theorem</b> below. Proofs are based on estimates for the entropy numbers of the compact trace operator </p><div><div><span>$$begin{aligned} textrm{tr},_mu : quad B^s_p ({{mathbb {R}}}^n) hookrightarrow L_p (Gamma , mu ), quad s&gt;0, quad 1&lt;p&lt;infty . end{aligned}$$</span></div></div><p>We add at the end of the paper a few personal reminiscences illuminating the role of Pietsch in connection with the creation of approximation numbers and entropy numbers.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00381-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142410823","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}
引用次数: 0
Hyponormal measurable operators, affiliated to a semifinite von Neumann algebra 超常可测算子，隶属于半有限冯-诺依曼代数
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-07 DOI: 10.1007/s43036-024-00388-9
Airat Bikchentaev
{"title":"Hyponormal measurable operators, affiliated to a semifinite von Neumann algebra","authors":"Airat Bikchentaev","doi":"10.1007/s43036-024-00388-9","DOIUrl":"10.1007/s43036-024-00388-9","url":null,"abstract":"<div><p>Let <span>(mathcal {M})</span> be a von Neumann algebra of operators on a Hilbert space <span>(mathcal {H})</span> and <span>(tau )</span> be a faithful normal semifinite trace on <span>(mathcal {M})</span>, <span>(S(mathcal {M}, tau ))</span> be the <span>( ^*)</span>-algebra of all <span>(tau )</span>-measurable operators. Assume that an operator <span>(Tin S(mathcal {M}, tau ))</span> is paranormal or <span>( ^*)</span>-paranormal. If <span>(T^n)</span> is <span>(tau )</span>-compact for some <span>(nin mathbb {N})</span> then <i>T</i> is <span>(tau )</span>-compact; if <span>(T^n=0)</span> for some <span>(nin mathbb {N})</span> then <span>(T=0)</span>; if <span>(T^3=T)</span> then <span>(T=T^*)</span>; if <span>(T^2in L_1(mathcal {M}, tau ))</span> then <span>(Tin L_2(mathcal {M}, tau ))</span> and <span>(Vert TVert _2^2=Vert T^2Vert _1)</span>. If an operator <span>(Tin S(mathcal {M}, tau ))</span> is hyponormal and <span>(T^{*p}T^q)</span> is <span>(tau )</span>-compact for some <span>(p, q in mathbb {N}cup {0})</span>, <span>(p+q ge 1)</span> then <i>T</i> is normal. If <span>(Tin S(mathcal {M}, tau ))</span> is <i>p</i>-hyponormal for some <span>(0&lt;ple 1)</span> then the operator <span>((T^*T)^p-(TT^*)^p)</span> cannot have the inverse in <span>( mathcal {M})</span>. If an operator <span>(Tin S(mathcal {M}, tau ))</span> is hyponormal (or cohyponormal) and the operator <span>(T^2)</span> is Hermitian then <i>T</i> is normal.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142410387","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}
引用次数: 0
The method of cyclic resolvents for quasi-convex functions and quasi-nonexpansive mappings 准凸函数和准无穷映射的循环解析子方法
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-07 DOI: 10.1007/s43036-024-00390-1
Hadi Khatibzadeh, Maryam Moosavi
{"title":"The method of cyclic resolvents for quasi-convex functions and quasi-nonexpansive mappings","authors":"Hadi Khatibzadeh,&nbsp;Maryam Moosavi","doi":"10.1007/s43036-024-00390-1","DOIUrl":"10.1007/s43036-024-00390-1","url":null,"abstract":"<div><p>The method of cyclic resolvents has been extended for a finite family of quasi-convex functions and quasi-nonexpansive mappings in Hadamard spaces. The essential tool for proving the main results is the use of the recent article by the first author and Mohebbi on the behavior of an iteration of a strongly quasi-nonexpansive sequence. The results are new even in Hilbert spaces.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142410414","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}
引用次数: 0
Inequalities between s-numbers s 数之间的不等式
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-05 DOI: 10.1007/s43036-024-00386-x
Mario Ullrich
{"title":"Inequalities between s-numbers","authors":"Mario Ullrich","doi":"10.1007/s43036-024-00386-x","DOIUrl":"10.1007/s43036-024-00386-x","url":null,"abstract":"<div><p>Singular numbers of linear operators between Hilbert spaces were generalized to Banach spaces by s-numbers (in the sense of Pietsch). This allows for different choices, including approximation, Gelfand, Kolmogorov and Bernstein numbers. Here, we present an elementary proof of a bound between the smallest and the largest s-number.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00386-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142410100","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}
引用次数: 0
Norm behavior of Jordan and bidiagonal matrices 约旦矩阵和对角线矩阵的规范行为
IF 0.8
Advances in Operator Theory Pub Date : 2024-09-23 DOI: 10.1007/s43036-024-00378-x
G. Krishna Kumar, P. V. Vivek
{"title":"Norm behavior of Jordan and bidiagonal matrices","authors":"G. Krishna Kumar,&nbsp;P. V. Vivek","doi":"10.1007/s43036-024-00378-x","DOIUrl":"10.1007/s43036-024-00378-x","url":null,"abstract":"<div><p>Determining the norm behavior of non-normal matrices from the sets related to the spectrum is one of the fundamental problems of matrix theory. This article proves that the pseudospectra and condition spectra determine the norm behavior of Jordan matrices for any matrix <i>p</i>-norm. Further, sufficient conditions for determining the 1-norm and infinity norm behavior of bidiagonal matrices from the pseudospectra and condition spectra are also provided.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413397","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}
引用次数: 0
Spectral reconstruction of operator tuples 算子元组的频谱重构
IF 0.8
Advances in Operator Theory Pub Date : 2024-09-10 DOI: 10.1007/s43036-024-00380-3
Michael I. Stessin
{"title":"Spectral reconstruction of operator tuples","authors":"Michael I. Stessin","doi":"10.1007/s43036-024-00380-3","DOIUrl":"10.1007/s43036-024-00380-3","url":null,"abstract":"<div><p>The spectral theorem implies that the spectrum of a bounded normal operator acting on a Hilbert space provides a substantial information about the operator. For example, the set of eigenvalues of a normal matrix and their respective multiplicities determine the matrix up to a unitary equivalence, while the spectral measure, <span>(E_B(lambda ))</span> of a normal operator <i>B</i> acting on a Hilbert space determines <i>B</i> via the integral spectral resolution, </p><div><div><span>$$begin{aligned} B=int _{sigma (B)} lambda dE_B(lambda ). end{aligned}$$</span></div></div><p>In general, for a non-normal operator the spectrum provides a rather limited information about the operator. In this paper we show that, if we include an arbitrary bounded operator <i>B</i> acting on a separable Hilbert space into a quadruple which contains 3 specific operators along with <i>B</i>, it is possible to reconstruct <i>B</i> from the proper projective joint spectrum of the quadruple (and here we mean reconstruct precisely, not up to an equivalence). We call this process <b>spectral reconstruction</b>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142411243","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}
引用次数: 0
Asymptotics of the eigenvalues of seven-diagonal Toeplitz matrices of a special form 特殊形式的七对角托普利兹矩阵特征值的渐近性
IF 0.8
Advances in Operator Theory Pub Date : 2024-09-04 DOI: 10.1007/s43036-024-00374-1
M. Barrera, S. Grudsky, V. Stukopin, I. Voronin
{"title":"Asymptotics of the eigenvalues of seven-diagonal Toeplitz matrices of a special form","authors":"M. Barrera,&nbsp;S. Grudsky,&nbsp;V. Stukopin,&nbsp;I. Voronin","doi":"10.1007/s43036-024-00374-1","DOIUrl":"10.1007/s43036-024-00374-1","url":null,"abstract":"<div><p>This work is devoted to the construction of a uniform asymptotics in the dimension of the matrix n tending to infinity of all eigenvalues in the case of a seven-diagonal Toeplitz matrix with a symbol having a zero of the sixth order, while the cases of symbols with zeros of the second and fourth orders were considered earlier. On the other hand, the results obtained refine the results of the classical work of Parter and Widom on the asymptotics of the extreme eigenvalues. We also note that the obtained formulas showed high computational efficiency both in sense of accuracy (already for relatively small values of n) and in sense of speed.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409895","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}
引用次数: 0
Embedding theorems for Besov–Morrey spaces 贝索夫-莫雷空间的嵌入定理
IF 0.8
Advances in Operator Theory Pub Date : 2024-08-22 DOI: 10.1007/s43036-024-00377-y
Arash Ghorbanalizadeh, Tahereh Khazaee
{"title":"Embedding theorems for Besov–Morrey spaces","authors":"Arash Ghorbanalizadeh,&nbsp;Tahereh Khazaee","doi":"10.1007/s43036-024-00377-y","DOIUrl":"10.1007/s43036-024-00377-y","url":null,"abstract":"<div><p>The purpose of this paper is to investigate the embedding theorems for Besov–Morrey spaces using the equivalence theorem for the <i>K</i>-functional and the modulus of continuity on Morrey spaces. First, we obtain some theorems in ball Banach function space and then focus on Morrey spaces. The Marchaud’s inequality on Morrey spaces and a specific case of embedding theorems for Sobolev–Morrey spaces are crucial tools. We show that the Besov–Morrey space <span>(B_{alpha , a}^{p,lambda }(mathbb {R}^{n}))</span> is continuously embedded in the Morrey-Lorentz space <span>(mathcal {M}_{q,p}^{lambda }(mathbb {R}^{n}))</span>, and also, for any <span>(alpha , beta &gt; 0)</span> and <span>(1&lt; ale p &lt; q le infty )</span>, the Besov–Morrey space <span>(B_{alpha + beta , a}^{p,lambda }(mathbb {R}^{n}))</span> is continuously embedded in the Besov–Morrey space <span>(B_{beta , a}^{q,lambda }(mathbb {R}^{n}))</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413066","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}
引用次数: 0
Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure 希尔伯特空间估值高斯过程及其核、因式分解和协方差结构
IF 0.8
Advances in Operator Theory Pub Date : 2024-08-19 DOI: 10.1007/s43036-024-00375-0
Palle E. T. Jorgensen, James Tian
{"title":"Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure","authors":"Palle E. T. Jorgensen,&nbsp;James Tian","doi":"10.1007/s43036-024-00375-0","DOIUrl":"10.1007/s43036-024-00375-0","url":null,"abstract":"<div><p>Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation constructions in operator theory, and the second to general classes of stochastic processes. For the latter, we apply our operator valued kernel-results in order to build new Hilbert space-valued Gaussian processes, and to analyze their structures of covariance configurations.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412366","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}
引用次数: 0
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