{"title":"Disjoint hypercyclic Toeplitz operators","authors":"Özkan Değer, Beyaz Başak Eskişehirli","doi":"10.1007/s00013-024-02084-9","DOIUrl":"10.1007/s00013-024-02084-9","url":null,"abstract":"<div><p>The aim of this work is to describe new classes of disjoint hypercyclic Toeplitz operators on the Hardy space <span>(H^2({mathbb {D}}))</span> in the unit disc <span>({mathbb {D}})</span>. We examine the disjoint hypercyclicity of the coanalytic Toeplitz operators, the Toeplitz operators with the symbols <span>(a{bar{z}}+b+cz)</span>, where <span>(a,b,cin {mathbb {C}})</span>, and the Toeplitz operators with the symbols <span>(p(bar{z})+varphi (z))</span>, where <i>p</i> is a polynomial and <span>(varphi in H^infty (mathbb {D}))</span>. The hypercyclicity of these classes of Toeplitz operators has been characterized by G. Godefroy and J. Shapiro (J. Funct. Anal., 98, 1991), S. Shkarin (arXiv:1210.3191v1, 2012), and A. Baranov and L. Lishanskii (Results Math., 70, 2016), respectively. Based on their results, we first provide a criterion for the bounded linear operators to be disjoint hypercyclic. Using this criterion, we then establish certain conditions under which the aforementioned classes of Toeplitz operators are disjoint hypercyclic in terms of their symbols.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"301 - 310"},"PeriodicalIF":0.5,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184795","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}
{"title":"New results on maximal (L^p)-regularity of a class of integrodifferential equations","authors":"H. Bounit, S. Hadd, Y. Manar","doi":"10.1007/s00013-024-02100-y","DOIUrl":"10.1007/s00013-024-02100-y","url":null,"abstract":"<div><p>The aim of this study is twofold. Initially, by employing a perturbation semigroup approach and admissible observation operators, a novel variation of constants formula is presented for the mild solutions of a specific set of integrodifferential equations in Banach spaces. Subsequently, utilizing this formula, an examination of the maximal <span>(L^p)</span>-regularity for such equations is conducted through the application of the sum operator method established by Da Prato and Grisvard. Importantly, it is demonstrated that the maximal <span>(L^p)</span>-regularity of an integrodifferential equation is equivalent to that of the same equation when the integral term is omitted. Furthermore, a finding concerning the strong solution of an initial value integrodifferential equation is provided when the initial condition pertains to the trace space.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"325 - 341"},"PeriodicalIF":0.5,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184641","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}
William Cason, Akash Jim, Charlie Medlock, Erick Ross, Hui Xue
{"title":"On the average size of the eigenvalues of the Hecke operators","authors":"William Cason, Akash Jim, Charlie Medlock, Erick Ross, Hui Xue","doi":"10.1007/s00013-024-02089-4","DOIUrl":"10.1007/s00013-024-02089-4","url":null,"abstract":"<div><p>We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space <span>(S_k(Gamma _0(N)))</span> in both the vertical and the horizontal perspective. The “average size” is measured via the quadratic mean.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"255 - 263"},"PeriodicalIF":0.5,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02089-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184870","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}
Lajos Hajdu, Florian Luca, Szabolcs Tengely, Maciej Ulas
{"title":"Products of Catalan numbers which are squares","authors":"Lajos Hajdu, Florian Luca, Szabolcs Tengely, Maciej Ulas","doi":"10.1007/s00013-024-02088-5","DOIUrl":"10.1007/s00013-024-02088-5","url":null,"abstract":"<div><p>Let <span>(C_{n})</span> be the <i>n</i>-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each <span>(kge 3)</span>, there are infinitely many <i>k</i>-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of <span>(xin mathbb {N}_{+})</span> such that <span>(C_{x}C_{x+1})</span> is a power-full number and prove that there are infinitely many such <i>x</i>. Moreover we present some numerical results which motivate further problems.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"265 - 281"},"PeriodicalIF":0.5,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184636","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}
{"title":"On the rank of projective modules","authors":"F.E.A. Johnson","doi":"10.1007/s00013-024-02081-y","DOIUrl":"10.1007/s00013-024-02081-y","url":null,"abstract":"<div><p>Let <i>P</i> be a nonzero projective module over an integral group ring. We consider the question of whether the rank of <i>P</i> is necessarily positive.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"233 - 241"},"PeriodicalIF":0.5,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02081-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184864","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}
{"title":"Correction to: A reciprocity law in function fields","authors":"Yoshinori Hamahata","doi":"10.1007/s00013-024-02092-9","DOIUrl":"10.1007/s00013-024-02092-9","url":null,"abstract":"","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"355 - 356"},"PeriodicalIF":0.5,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02092-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184800","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}
M. M. Chems-Eddin, B. Feryouch, H. Mouanis, A. Tamoussit
{"title":"On the Krull dimension of rings of integer-valued rational functions","authors":"M. M. Chems-Eddin, B. Feryouch, H. Mouanis, A. Tamoussit","doi":"10.1007/s00013-024-02086-7","DOIUrl":"10.1007/s00013-024-02086-7","url":null,"abstract":"<div><p>Let <i>D</i> be an integral domain with quotient field <i>K</i> and <i>E</i> a subset of <i>K</i>. The <i>ring of integer-valued rational functions on</i> <i>E</i> is defined as </p><div><div><span>$$begin{aligned} mathrm {Int^R}(E,D):=lbrace varphi in K(X);; varphi (E)subseteq Drbrace . end{aligned}$$</span></div></div><p>The main goal of this paper is to investigate the Krull dimension of the ring <span>(mathrm {Int^R}(E,D).)</span> Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"243 - 254"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184674","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}
{"title":"Choquet integrals, Hausdorff content and sparse operators","authors":"Naoya Hatano, Ryota Kawasumi, Hiroki Saito, Hitoshi Tanaka","doi":"10.1007/s00013-024-02083-w","DOIUrl":"10.1007/s00013-024-02083-w","url":null,"abstract":"<div><p>Let <span>(H^d)</span>, <span>(0<d<n)</span>, be the dyadic Hausdorff content of the <i>n</i>-dimensional Euclidean space <span>({{mathbb {R}}}^n)</span>. It is shown that <span>(H^d)</span> counts a Cantor set of the unit cube <span>([0, 1)^n)</span> as <span>(approx 1)</span>, which implies the unboundedness of the sparse operator <span>({{mathcal {A}}}_{{{mathcal {S}}}})</span> on the Choquet space <span>({mathcal L}^p(H^d))</span>, <span>(p>0)</span>. In this paper, the sparse operator <span>({mathcal A}_{{{mathcal {S}}}})</span> is proved to map <span>({{mathcal {L}}}^p(H^d))</span>, <span>(1le p<infty )</span>, into an associate space of the Orlicz-Morrey space <span>({{{mathcal {M}}}^{p'}_{Phi _0}(H^d)}')</span>, <span>(Phi _0(t)=tlog (e+t))</span>. Further, another characterization of those associate spaces is given by means of the tiling <span>({{mathcal {T}}})</span> of <span>({{mathbb {R}}}^n)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"311 - 324"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184676","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}
{"title":"A seminorm characterization of infinite Banach direct sums","authors":"Hojjatollah Samea","doi":"10.1007/s00013-024-02080-z","DOIUrl":"10.1007/s00013-024-02080-z","url":null,"abstract":"<div><p>In this paper, the notion of a <span>(Delta )</span>-direct sum of a family of Banach spaces indexed by a set <i>I</i>, where <span>(Delta )</span> is a union-closed subnet of <span>(textsf{Fin}(I))</span> (the family of all finite subsets of <i>I</i>), is introduced. A seminorm characterization of <span>(Delta )</span>-direct sums and some results are presented. Necessary and sufficient conditions are found that a direct sum of a family of Banach spaces is a <span>(Delta )</span>-direct sum. Elements of a direct sum of Banach spaces that are <span>(Delta )</span>-sectionally convergent are introduced and studied. Examples of <span>(Delta )</span>-direct sums and applications of <span>(Delta )</span>-direct sums to Fourier analysis on compact groups are given.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"283 - 299"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184675","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}
{"title":"Linear maps preserving the inclusion of fixed subsets into the local spectrum at some fixed vector","authors":"Constantin Costara","doi":"10.1007/s00013-024-02078-7","DOIUrl":"10.1007/s00013-024-02078-7","url":null,"abstract":"<div><p>For a natural number <span>(n ge 2)</span>, denote by <span>(mathcal {M}_{n})</span> the space of all <span>(ntimes n)</span> matrices over the complex field <span>(mathbb {C})</span>. Let <span>(x_0 in mathbb {C}^{n})</span> be a fixed nonzero vector, and fix also two nonempty subsets <span>(K_1, K_2 subseteq mathbb {C})</span>, each having at most <i>n</i> distinct elements. Under the assumption that <span>(|K_1| le |K_2|)</span>, we characterize linear bijective maps <span>(varphi )</span> on <span>(mathcal {M}_{n})</span> having the property that, for each matrix <i>T</i>, we have that <span>(K_2)</span> is a subset of the local spectrum of <span>(varphi (T))</span> at <span>(x_0 )</span> whenever <span>(K_1 )</span> is a subset of the local spectrum of <i>T</i> at <span>(x_0)</span>. As a corollary, we also characterize linear maps <span>(varphi )</span> on <span>(mathcal {M} _{n})</span> having the property that, for each matrix <i>T</i>, we have that <span>(K_1)</span> is a subset of the local spectrum of <i>T</i> at <span>(x_0)</span> if and only if <span>(K_2)</span> is a subset of the local spectrum of <span>(varphi (T))</span> at <span>(x_0)</span>, without the bijectivity assumption on the map <span>(varphi )</span> and with no assumption made regarding the number of elements of <span>(K_1)</span> and <span>(K_2)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 2","pages":"165 - 176"},"PeriodicalIF":0.5,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02078-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108624","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}
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