The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BO-TI], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a right-invariant metric carrying flows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closed-graph theorem, to the Birkhoff-Kakutani metrization theorem and further back still to Banach's Theorie des operations lineaires. Its most recent noteworthy appearance has been in connection with the Effros Open Mapping Principle. We collect together known salient features and develop their theory including Steinhaus theory unified by the Category Embedding Theorem [BO-LBII], the associated themes of subadditivity and convexity, and a topological duality inherent to topological dynamics. We study the latter both for its independent interest and as a foundation for topological regular variation.
{"title":"Normed versus topological groups: dichotomy and duality","authors":"N. Bingham, A. Ostaszewski","doi":"10.4064/DM472-0-1","DOIUrl":"https://doi.org/10.4064/DM472-0-1","url":null,"abstract":"The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BO-TI], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a right-invariant metric carrying flows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closed-graph theorem, to the Birkhoff-Kakutani metrization theorem and further back still to Banach's Theorie des operations lineaires. Its most recent noteworthy appearance has been in connection with the Effros Open Mapping Principle. We collect together known salient features and develop their theory including Steinhaus theory unified by the Category Embedding Theorem [BO-LBII], the associated themes of subadditivity and convexity, and a topological duality inherent to topological dynamics. We study the latter both for its independent interest and as a foundation for topological regular variation.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"472 1","pages":"1-138"},"PeriodicalIF":1.8,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156170","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}
The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for scalar valued and matrix valued Hamiltonians. In the semiclassical limit h → 0 we consider resonances near the real axis and we establish the CAP method rigorously in an abstract matrix valued setting by proving that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proof is based on pseudodifferential operator theory and microlocal analysis.
{"title":"Complex absorbing potential method for systems","authors":"J. Kungsman, M. Melgaard","doi":"10.4064/DM469-0-1","DOIUrl":"https://doi.org/10.4064/DM469-0-1","url":null,"abstract":"The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for scalar valued and matrix valued Hamiltonians. In the semiclassical limit h → 0 we consider resonances near the real axis and we establish the CAP method rigorously in an abstract matrix valued setting by proving that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proof is based on pseudodifferential operator theory and microlocal analysis.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"469 1","pages":"1-58"},"PeriodicalIF":1.8,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156105","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}
In recent years, there have been several studies of various `approximate' versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004. The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability. There are a number of open questions on these relationships; these will be considered. In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work; we shall concentrate on the case of Banach function algebras. In Chapter 2, we shall discuss these properties for the semigroup algebra `1(S) of a semigroup S. In the case where S has only finitely many idempotents, `1(S) is approximately amenable if and only if it is amenable. In Chapter 3, we shall consider the class of weighted semigroup algebras of the form `1(N^; !), where ! : Z ! [1; 1) is an arbitrary function. We shall determine necessary and sufficient conditions on ! for these Banach sequence algebras to have each of the various approximate amenability properties that interest us. In this way we shall illuminate the implications between these properties. In Chapter 4, we shall discuss Segal algebras on T and on R. It is a conjecture that every proper Segal algebra on T fails to be approximately amenable; we shall establish this conjecture for a wide class of Segal algebras.
{"title":"Approximate amenability of semigroup algebras and Segal algebras","authors":"H. Dales, R. J. Loy","doi":"10.4064/DM474-0-1","DOIUrl":"https://doi.org/10.4064/DM474-0-1","url":null,"abstract":"In recent years, there have been several studies of various `approximate' versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004. The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability. There are a number of open questions on these relationships; these will be considered. In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work; we shall concentrate on the case of Banach function algebras. In Chapter 2, we shall discuss these properties for the semigroup algebra `1(S) of a semigroup S. In the case where S has only finitely many idempotents, `1(S) is approximately amenable if and only if it is amenable. In Chapter 3, we shall consider the class of weighted semigroup algebras of the form `1(N^; !), where ! : Z ! [1; 1) is an arbitrary function. We shall determine necessary and sufficient conditions on ! for these Banach sequence algebras to have each of the various approximate amenability properties that interest us. In this way we shall illuminate the implications between these properties. In Chapter 4, we shall discuss Segal algebras on T and on R. It is a conjecture that every proper Segal algebra on T fails to be approximately amenable; we shall establish this conjecture for a wide class of Segal algebras.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"37 1","pages":"1-58"},"PeriodicalIF":1.8,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156222","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}
We prove weighted norm inequalities of integral type between the gradients of solutions u of elliptic equations and their boundary data f on bounded Lipschitz domains. 0. Introduction. We are interested in the following general question: To what extent is the interior smoothness of the solution of a PDE controlled by the size of its boundary values? To be more specific, suppose (for now) that Ω ⊂ R is a nice domain, μ is a positive measure supported in Ω, and v is a non-negative measurable function defined on ∂Ω. If f : ∂Ω → R is reasonable (say, continuous function with compact support), we let u : Ω → R be the solution of the classical Dirichlet problem with boundary values equal to f . (We are implicitly assuming that Ω is nice enough to have this make sense!) Let p and q be real numbers lying strictly between 1 and infinity. When is it the case that (∫ Ω |∇u|q dμ )1/q ≤ (∫
{"title":"Weighted inequalities for gradients on non-smooth domains","authors":"C. Sweezy, J. Wilson","doi":"10.4064/DM471-0-1","DOIUrl":"https://doi.org/10.4064/DM471-0-1","url":null,"abstract":"We prove weighted norm inequalities of integral type between the gradients of solutions u of elliptic equations and their boundary data f on bounded Lipschitz domains. 0. Introduction. We are interested in the following general question: To what extent is the interior smoothness of the solution of a PDE controlled by the size of its boundary values? To be more specific, suppose (for now) that Ω ⊂ R is a nice domain, μ is a positive measure supported in Ω, and v is a non-negative measurable function defined on ∂Ω. If f : ∂Ω → R is reasonable (say, continuous function with compact support), we let u : Ω → R be the solution of the classical Dirichlet problem with boundary values equal to f . (We are implicitly assuming that Ω is nice enough to have this make sense!) Let p and q be real numbers lying strictly between 1 and infinity. When is it the case that (∫ Ω |∇u|q dμ )1/q ≤ (∫","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"471 1","pages":"1-53"},"PeriodicalIF":1.8,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156152","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}
Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $sH$ with a multivalued part $mul A$. An operator $B$ in $sH$ with $ran Bperpmul A^{**}$ is said to be an operator part of $A$ when $A=B hplus ({0}times mul A)$, where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if $A=U+I V$, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated.
{"title":"Componentwise and Cartesian decompositions of linear relations","authors":"S. Hassi, H. D. Snoo, F. H. Szafraniec","doi":"10.4064/DM465-0-1","DOIUrl":"https://doi.org/10.4064/DM465-0-1","url":null,"abstract":"Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $sH$ with a multivalued part $mul A$. An operator $B$ in $sH$ with $ran Bperpmul A^{**}$ is said to be an operator part of $A$ when $A=B hplus ({0}times mul A)$, where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if $A=U+I V$, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"465 1","pages":"4-58"},"PeriodicalIF":1.8,"publicationDate":"2009-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156278","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}
{"title":"Refined Kodaira classes and conductors of twisted elliptic curves","authors":"J. Browkin, Daniel Davies","doi":"10.4064/DM463-0-1","DOIUrl":"https://doi.org/10.4064/DM463-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"463 1","pages":"1-45"},"PeriodicalIF":1.8,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156188","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}
G. Fremiot, W. Horn, A. Laurain, Murali Rao, J. Sokołowski
{"title":"On the analysis of boundary value problems in nonsmooth domains","authors":"G. Fremiot, W. Horn, A. Laurain, Murali Rao, J. Sokołowski","doi":"10.4064/DM462-0-1","DOIUrl":"https://doi.org/10.4064/DM462-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"462 1","pages":"1-149"},"PeriodicalIF":1.8,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156178","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}
{"title":"Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces","authors":"A. Caetano, Sofia Lopes","doi":"10.4064/DM460-0-1","DOIUrl":"https://doi.org/10.4064/DM460-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"460 1","pages":"1-44"},"PeriodicalIF":1.8,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156139","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}
{"title":"Weighted convolution algebras on subsemigroups of the real line","authors":"H. Dales, H. V. Dedania","doi":"10.4064/DM459-0-1","DOIUrl":"https://doi.org/10.4064/DM459-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"38 1","pages":"1-60"},"PeriodicalIF":1.8,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4064/DM459-0-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156103","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}
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