Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0226-2
S. Dehimi, M. H. Mortad, A. Bachir
In this paper, we mainly show that if a product AB (or BA) of a closed symmetric operator A and a bounded positive operator B is normal, then it is self-adjoint. Equivalently, this means that B commutes with A. Certain generalizations and consequences are also presented.
{"title":"On the Commutativity of Closed Symmetric Operators","authors":"S. Dehimi, M. H. Mortad, A. Bachir","doi":"10.1007/s10476-023-0226-2","DOIUrl":"10.1007/s10476-023-0226-2","url":null,"abstract":"<div><p>In this paper, we mainly show that if a product <i>AB</i> (or <i>BA</i>) of a closed symmetric operator <i>A</i> and a bounded positive operator <i>B</i> is normal, then it is self-adjoint. Equivalently, this means that <i>B</i> commutes with <i>A</i>. Certain generalizations and consequences are also presented.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43472144","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}
Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0229-z
A. Ge, Q. He, D. Yan
Let ({cal L} = - Delta + V) be a Schrödinger operator with a nonnegative potential V belonging to the reverse Hölder class Bq for q> n/2. In this paper, we study the weighted compactness of oscillation and variation commutators generated by BMO-type functions and some Schrödinger operators, which include Riesz transform and other standard Calderón–Zygmund operators.
{"title":"On Weighted Compactness of Oscillation and Variation of Commutators Associated with Schrödinger Operators","authors":"A. Ge, Q. He, D. Yan","doi":"10.1007/s10476-023-0229-z","DOIUrl":"10.1007/s10476-023-0229-z","url":null,"abstract":"<div><p>Let <span>({cal L} = - Delta + V)</span> be a Schrödinger operator with a nonnegative potential <i>V</i> belonging to the reverse Hölder class <i>B</i><sub><i>q</i></sub> for <i>q</i>> <i>n</i>/2. In this paper, we study the weighted compactness of oscillation and variation commutators generated by BMO-type functions and some Schrödinger operators, which include Riesz transform and other standard Calderón–Zygmund operators.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10476-023-0229-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41861009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0225-3
J.-F. Chen, Y.-Y. Feng
By utilizing Nevanlinna theory of meromorphic functions, we characterize meromorphic solutions of the following nonlinear differential equation of the form
where n ≥ 3, t ≥ 0 and m ≥ 1 are integers, n ≥ m, P(z, f, f′, …, f(t)) is a differential polynomial in f (z) of degree d ≤ n with small functions of f (z) as its coefficients, and αj, Pj (j = 1, 2, …, m) are nonzero constants such that ∣α1∣ > ∣α2∣ > … > ∣αm∣. Also we provide the concrete forms of the solutions of the equation above, and present some examples illustrating the sharpness of our results.
{"title":"On Meromorphic Solutions of Nonlinear Complex Differential Equations","authors":"J.-F. Chen, Y.-Y. Feng","doi":"10.1007/s10476-023-0225-3","DOIUrl":"10.1007/s10476-023-0225-3","url":null,"abstract":"<div><p>By utilizing Nevanlinna theory of meromorphic functions, we characterize meromorphic solutions of the following nonlinear differential equation of the form </p><div><div><span>$${f^n}{f^prime } + P(z,f,{f^prime }, ldots ,{f^{(t)}}) = {P_1}{e^{{alpha _1}z}} + {P_2}{e^{{alpha _2}z}} + cdots + {P_m}{e^{{alpha _m}z}},$$</span></div></div><p> where <i>n</i> ≥ 3, <i>t</i> ≥ 0 and <i>m</i> ≥ 1 are integers, <i>n</i> ≥ <i>m, P</i>(<i>z, f, f′, …, f</i><sup>(<i>t</i>)</sup>) is a differential polynomial in <i>f</i> (<i>z</i>) of degree <i>d</i> ≤ <i>n</i> with small functions of <i>f</i> (<i>z</i>) as its coefficients, and α<sub><i>j</i></sub>, <i>P</i><sub><i>j</i></sub> (<i>j</i> = 1, 2, …, <i>m</i>) are nonzero constants such that ∣α<sub>1</sub>∣ > ∣α<sub>2</sub>∣ > … > ∣α<sub><i>m</i></sub>∣. Also we provide the concrete forms of the solutions of the equation above, and present some examples illustrating the sharpness of our results.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10476-023-0225-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41583690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0234-2
S. Soltani Renani, Z. Yari
Let G be a locally compact group, ({cal B}({L^2}(G))) be the space of all bounded linear operators on L2(G), and (({cal T}({L^2}(G)), ast)) be the Banach algebra of trace class operators on L2(G). In this paper, we focus on some Banach right submodules of ({cal B}({L^2}(G))) over the convolution algebras (({cal T}({L^2}(G)), ast)) and (L1(G),*). We will see that if the locally compact group G is discrete, then the Banach right ℓ1(G)-module structures of them are derived from their Banach right ({cal T}({ell ^2}(G)))-module structures. We also study the projectivity of these Banach right ℓ1(G)-modules.
{"title":"Projectivity of Some Banach Right Modules over the Group Algebra ℓ1(G)","authors":"S. Soltani Renani, Z. Yari","doi":"10.1007/s10476-023-0234-2","DOIUrl":"10.1007/s10476-023-0234-2","url":null,"abstract":"<div><p>Let <i>G</i> be a locally compact group, <span>({cal B}({L^2}(G)))</span> be the space of all bounded linear operators on <i>L</i><sup>2</sup>(<i>G</i>), and <span>(({cal T}({L^2}(G)), ast))</span> be the Banach algebra of trace class operators on <i>L</i><sup>2</sup>(<i>G</i>). In this paper, we focus on some Banach right submodules of <span>({cal B}({L^2}(G)))</span> over the convolution algebras <span>(({cal T}({L^2}(G)), ast))</span> and (<i>L</i><sup>1</sup>(<i>G</i>),*). We will see that if the locally compact group <i>G</i> is discrete, then the Banach right <i>ℓ</i><sup>1</sup>(<i>G</i>)-module structures of them are derived from their Banach right <span>({cal T}({ell ^2}(G)))</span>-module structures. We also study the projectivity of these Banach right <i>ℓ</i><sup>1</sup>(<i>G</i>)-modules.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46322216","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}
Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0233-3
A. G. Smirnov, M. S. Smirnov
We propose a unified treatment of extensions of group-valued contents (i.e., additive set functions defined on a ring) by means of adding new null sets. Our approach is based on the notion of a completion ring for a content μ. With every such ring ({cal N}), an extension of μ is naturally associated which is called the ({cal N})-completion of μ. The ({cal N})-completion operation comprises most previously known completion-type procedures and also gives rise to some new extensions, which may be useful for constructing counterexamples in measure theory. We find a condition ensuring that σ-additivity of a content is preserved under the ({cal N})-completion and establish a criterion for the ({cal N})-completion of a measure to be again a measure.
{"title":"Completion Procedures in Measure Theory","authors":"A. G. Smirnov, M. S. Smirnov","doi":"10.1007/s10476-023-0233-3","DOIUrl":"10.1007/s10476-023-0233-3","url":null,"abstract":"<div><p>We propose a unified treatment of extensions of group-valued contents (i.e., additive set functions defined on a ring) by means of adding new null sets. Our approach is based on the notion of a completion ring for a content <i>μ</i>. With every such ring <span>({cal N})</span>, an extension of <i>μ</i> is naturally associated which is called the <span>({cal N})</span>-completion of <i>μ</i>. The <span>({cal N})</span>-completion operation comprises most previously known completion-type procedures and also gives rise to some new extensions, which may be useful for constructing counterexamples in measure theory. We find a condition ensuring that <i>σ</i>-additivity of a content is preserved under the <span>({cal N})</span>-completion and establish a criterion for the <span>({cal N})</span>-completion of a measure to be again a measure.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42483533","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}
Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0231-5
G. Ramesh, H. Osaka, Y. Udagawa, T. Yamazaki
In this note we discuss absolutely norm attaining property (({cal A}{cal N})-property in short) of the Jordan product and Lie-bracket. We propose a functional calculus for positive absolutely norm attaining operators and discuss the stability of the ({cal A}{cal N})-property under the functional calculus. As a consequence we discuss the operator mean of positive ({cal A}{cal N})-operators.
{"title":"Stability of ({cal A}{cal N})-Operators under Functional Calculus","authors":"G. Ramesh, H. Osaka, Y. Udagawa, T. Yamazaki","doi":"10.1007/s10476-023-0231-5","DOIUrl":"10.1007/s10476-023-0231-5","url":null,"abstract":"<div><p>In this note we discuss absolutely norm attaining property (<span>({cal A}{cal N})</span>-property in short) of the Jordan product and Lie-bracket. We propose a functional calculus for positive absolutely norm attaining operators and discuss the stability of the <span>({cal A}{cal N})</span>-property under the functional calculus. As a consequence we discuss the operator mean of positive <span>({cal A}{cal N})</span>-operators.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10476-023-0231-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50431404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-06DOI: 10.1007/s10476-023-0230-6
A. Kowalski, I. I. Marchenko
This paper is devoted to the development of Beckenbach’s theory of the meromorphic minimal surfaces. We consider the relationship between the number of separated maximum points of a meromorphic minimal surface and the Baernstein’s T*-function. The results of Edrei, Goldberg, Heins, Ostrovskii, Wiman are generalized. We also give examples showing that the obtained estimates are sharp.
{"title":"On the Edrei–Goldberg–Ostrovskii Theorem for Minimal Surfaces","authors":"A. Kowalski, I. I. Marchenko","doi":"10.1007/s10476-023-0230-6","DOIUrl":"10.1007/s10476-023-0230-6","url":null,"abstract":"<div><p>This paper is devoted to the development of Beckenbach’s theory of the meromorphic minimal surfaces. We consider the relationship between the number of separated maximum points of a meromorphic minimal surface and the Baernstein’s <i>T</i>*-function. The results of Edrei, Goldberg, Heins, Ostrovskii, Wiman are generalized. We also give examples showing that the obtained estimates are sharp.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44302490","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}