Eduardo Brandani da Silva, Luan Carlos Della Pasqua
{"title":"Generalized entropy numbers of sets and operators","authors":"Eduardo Brandani da Silva, Luan Carlos Della Pasqua","doi":"10.1007/s43036-025-00474-6","DOIUrl":"10.1007/s43036-025-00474-6","url":null,"abstract":"<div><p>In this work, we introduce and study generalized entropy numbers for sets and operators acting on Banach spaces. The classical notion of Hausdorff entropy numbers becomes a particular case of the given definition. We also provide several other examples of generalized entropy numbers for sets and operators. We prove several properties for the general case.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145168633","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":"Notes on non-compact maps and the importance of Bernstein numbers","authors":"David E. Edmunds, Jan Lang","doi":"10.1007/s43036-025-00456-8","DOIUrl":"10.1007/s43036-025-00456-8","url":null,"abstract":"<div><p>In this review paper we study non-compact operators and embeddings between function spaces, highlighting interesting phenomena and the significance of Bernstein numbers. In particular, we demonstrate that for non-compact maps the usual <i>s</i>-numbers (e.g., approximation, Kolmogorov, and entropy numbers) fail to reveal finer structural properties, and one must instead consider concepts such as strict singularity and Bernstein numbers.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145073980","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":"Linearization and Lemma of Newton for operator functions","authors":"Matthias Stiefenhofer","doi":"10.1007/s43036-025-00472-8","DOIUrl":"10.1007/s43036-025-00472-8","url":null,"abstract":"<div><p>We study the action of the nonlinear mapping <i>G</i>[<i>z</i>] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of <span>(G[z]=0)</span>. The results represent local generalizations of the standard implicit or inverse function theorem and of Newton’s Lemma, considering the order of approximation needed to obtain solutions of <span>(G[z]=0)</span>. The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2<i>k</i> with <i>k</i> denoting the maximal rank of Jordan chains needed to ensure <i>k</i>-surjectivity of the linear family. Along these lines, the zero set of <i>G</i>[<i>z</i>] in the cone is calculated immediately, agreeing up to the order of <span>(k-1)</span> with the given approximation. Hence, the results may also be interpreted as a version of Tougeron’s implicit function theorem in Banach spaces, essentially restricted to the arc case of a single variable. Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-025-00472-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144934611","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":"Maps preserving the dimension of fixed points of Jordan triple product of operators","authors":"Rixia Song, Weijuan Shi","doi":"10.1007/s43036-025-00473-7","DOIUrl":"10.1007/s43036-025-00473-7","url":null,"abstract":"<div><p>Let <span>({mathcal {X}})</span> be a complex Banach space, and let <span>({mathcal {B}}({mathcal {X}}))</span> be the algebra of all bounded linear operators on <span>({mathcal {X}}.)</span> In this paper, we characterize the general forms of surjective maps on <span>({mathcal {B}}({mathcal {X}}))</span> that preserve the dimension of fixed points of Jordan triple product of operators.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144909834","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":"Isometries between spaces of metrics","authors":"Katsuhisa Koshino","doi":"10.1007/s43036-025-00471-9","DOIUrl":"10.1007/s43036-025-00471-9","url":null,"abstract":"<div><p>Given a metrizable space <i>Z</i>, denote by <span>(operatorname {PM}(Z))</span> the space of continuous bounded pseudometrics on <i>Z</i>, and denote by <span>(operatorname {AM}(Z))</span> the one of continuous bounded admissible metrics on <i>Z</i>, both of which are equipped with the sup-norm <span>(Vert cdot Vert .)</span> In this paper, we shall prove Banach–Stone type theorems on spaces of metrics, that is, for metrizable spaces <i>X</i> and <i>Y</i>, <i>X</i> and <i>Y</i> are homeomorphic if and only if there exists a surjective isometry <span>(T: operatorname {PM}(X) rightarrow operatorname {PM}(Y))</span> <span>((T: operatorname {AM}(X) rightarrow operatorname {AM}(Y)))</span> satisfying some conditions. Then for each surjective isometry <i>T</i>, there is a homeomorphism <span>(phi : Y rightarrow X)</span> such that for any <span>(d in operatorname {PM}(X))</span> and for any <span>(x, y in Y,)</span> <span>(T(d)(x,y) = d(phi (x),phi (y)).)</span> Except for the case where the cardinality of <i>X</i> or <i>Y</i> is equal to 2, the homeomorphism <span>(phi )</span> can be chosen uniquely.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144843284","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":"Phase and norm retrievable operator valued frames","authors":"Dongwei Li, Yuxiang Xu","doi":"10.1007/s43036-025-00449-7","DOIUrl":"10.1007/s43036-025-00449-7","url":null,"abstract":"<div><p>In this paper, we investigate operator-valued frames (OPV-frames) for phase (norm) retrieval. Firstly, we give a sufficient and necessary condition for phase retrievable OPV-frames in real finite-dimensional Hilbert spaces. Some conditions which are equivalent to phase retrievable OPV-frames are also presented. Secondly, we obtain some equivalent conditions to the norm retrievable OPV-frame in real and complex finite-dimensional Hilbert spaces. Finally, we show that the property of phase retrievable for real Hilbert spaces is stable under small perturbation of an OPV-frame. It is also shown that the property of norm retrievability is stable under enough small perturbations of an OPV-frame only for phase retrievable OPV-frames.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145161475","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":"Strongly singular Calderón–Zygmund operators and commutators on Musielak–Orlicz Hardy spaces","authors":"Yanyan Han, Jinghan Shao, Huoxiong Wu","doi":"10.1007/s43036-025-00466-6","DOIUrl":"10.1007/s43036-025-00466-6","url":null,"abstract":"<div><p>This paper is devoted to studying the behaviors of strongly singular Calderón–Zygmund operators <i>T</i> and their commutators [<i>b</i>, <i>T</i>] generated by <i>T</i> with <span>(bin L_{loc}({mathbb {R}}^n))</span> on the Musielak–Orlicz Hardy spaces. The authors obtain the boundedness of <i>T</i> from the Musielak–Orlicz Hardy spaces <span>(H^varphi ({mathbb {R}}^n))</span> to the Musielak–Orlicz spaces <span>(L^varphi ({mathbb {R}}^n),)</span> and from the Musielak–Orlicz Hardy spaces <span>(H^varphi ({mathbb {R}}^n))</span> to themselves if <span>(T^*1=0.)</span> Meanwhile, the corresponding mapping properties for the commutators [<i>b</i>, <i>T</i>] are also obtained, provided that <i>b</i> belongs to <span>(mathcal {BMO}_{varphi ,u}({mathbb {R}}^n),)</span> a non-trivial subspace of <span>({textrm{BMO}}({mathbb {R}}^n).)</span></p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145161476","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}
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, Kasemsun Rutamorn
{"title":"Spectral theory for non-full commutative C*-categories","authors":"Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, Kasemsun Rutamorn","doi":"10.1007/s43036-025-00469-3","DOIUrl":"10.1007/s43036-025-00469-3","url":null,"abstract":"<div><p>We extend the spectral theory of commutative C*-categories to the non-full case, introducing a suitable notion of spectral spaceoid providing a duality between a category of “non-trivial” <span>(*)</span>-functors of non-full commutative C*-categories and a category of Takahashi morphisms of “non-full spaceoids” (here defined). As a byproduct we obtain a spectral theorem for a non-full generalization of imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras via continuous sections vanishing at infinity of a Hilbert C*-line-bundle over the graph of a homeomorphism between open subsets of the corresponding Gel’fand spectra of the C*-algebras.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145161116","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":"Framelets and wavelets with mixed dilation factors","authors":"Ran Lu","doi":"10.1007/s43036-025-00468-4","DOIUrl":"10.1007/s43036-025-00468-4","url":null,"abstract":"<div><p>As a main research area in applied and computational harmonic analysis, the theory and applications of framelets have been extensively investigated. Most existing literature is devoted to framelet systems that only use one dilation matrix as the sampling factor. To keep some key properties such as directionality, a framelet system often has a high redundancy rate. To reduce redundancy, a one-dimensional tight framelet with mixed dilation factors has been introduced for image processing. Though such tight framelets offer good performance in practice, their theoretical properties are far from being well understood. In this paper, we will systematically investigate framelets with mixed dilation factors, with arbitrary multiplicity in arbitrary dimensions. We will first study the discrete framelet transform employing a filter bank with mixed dilation factors and discuss its various properties. Next, we will introduce the notion of a discrete affine system in <span>(l_{2}(mathbb {Z}^d))</span> and study discrete framelet transforms with mixed dilation factors. Finally, we will discuss framelets and wavelets with mixed dilation factors in the space <span>(L_{2}(mathbb {R}^d))</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145170392","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":"Mixed product differences of composition operators and Volterra operators on Bloch spaces","authors":"Xin He, Cezhong Tong, Zicong Yang","doi":"10.1007/s43036-025-00470-w","DOIUrl":"10.1007/s43036-025-00470-w","url":null,"abstract":"<div><p>The differences of integration-composition operators on various analytic function spaces have attracted lots of attention for decades. In this note, we study the differences of mixed products of Volterra operators and composition operators on Bloch spaces. To be specific, we characterize the following four types of differences of mixed products: <span>(I_gC_{varphi }-C_{psi }J_h)</span>, <span>(J_gC_{varphi }-C_{psi }I_h)</span>, <span>(I_gC_{varphi }-C_{psi }I_h)</span> and <span>(J_gC_{varphi }-C_{psi }J_h)</span>. One surprising result is that unbounded <span>(I_gC_{varphi })</span> and <span>(C_{psi }J_h)</span> can not induce bounded difference <span>(I_gC_{varphi }-C_{psi }J_h)</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145169259","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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