Complex Analysis and Spectral Theory最新文献

{"title":"A panorama of positivity. II: Fixed\u0000 dimension","authors":"A. Belton, D. Guillot, A. Khare, M. Putinar","doi":"10.1090/conm/743/14958","DOIUrl":"https://doi.org/10.1090/conm/743/14958","url":null,"abstract":"This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.","PeriodicalId":213844,"journal":{"name":"Complex Analysis and Spectral Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128438327","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":"A holomorphic functional calculus for finite\u0000 families of commuting semigroups","authors":"J. Esterle","doi":"10.1090/conm/743/14956","DOIUrl":"https://doi.org/10.1090/conm/743/14956","url":null,"abstract":"Let A be a commutative Banach algebra such that uA = {0} for u $in$ A {0} which possesses dense principal ideals. The purpose of the paper is to give a general framework to define F (--$lambda$1$Delta$T 1 ,. .. , --$lambda$ k $Delta$T k) where F belongs to a natural class of holomorphic functions defined on suitable open subsets of C k containing the \"Arveson spectrum\" of (--$lambda$1$Delta$T 1 ,. .. , --$lambda$ k $Delta$T k), where $Delta$T 1 ,. .. , $Delta$T k are the infinitesimal generators of commuting one-parameter semigroups of multipliers on A belonging to one of the following classes (1) The class of strongly continous semigroups T = (T (te ia)t>0 such that $cup$t>0T (te ia)A is dense in A, where a $in$ R. (2) The class of semigroups T = (T ($zeta$)) $zeta$$in$S a,b holomorphic on an open sector S a,b such that T ($zeta$)A is dense in A for some, or equivalently for all $zeta$ $in$ S a,b. We use the notion of quasimultiplier, introduced in 1981 by the author at the Long Beach Conference on Banach algebras: the generators of the semigroups under consideration will be defined as quasimultipliers on A, and for $zeta$ in the Arveson resolvent set $sigma$ar($Delta$T) the resolvent ($Delta$T -- $zeta$I) --1 will be defined as a regular quasimultiplier on A, i.e. a quasimultiplier S on A such that sup n$ge$1 $lambda$ n S n u 0 and some u generating a dense ideal of A and belonging to the intersection of the domains of S n , n $ge$ 1. The first step consists in \"normalizing\" the Banach algebra A, i.e. continuously embedding A in a Banach algebra B having the same quasi-multiplier algebra as A but for which lim sup t$rightarrow$0 + T (te ia) M(B) < +$infty$ if T belongs to the class (1), and for which lim sup $zeta$$rightarrow$0 $zeta$$in$S $alpha$,$beta$ T ($zeta$) < +$infty$ for all pairs ($alpha$, $beta$) such that a < $alpha$ < $beta$ < b if T belongs to the class (2). Iterating this procedure this allows to consider ($lambda$j$Delta$T j + $zeta$I) --1 as an element of M(B) for $zeta$ $in$ Resar(--$lambda$j$Delta$T j), the \"Arveson resolvent set \" of --$lambda$j$Delta$T j , and to use the standard integral 'resolvent formula' even if the given semigroups are not bounded near the origin. A first approach to the functional calculus involves the dual G a,b of an algebra of fast decreasing functions, described in Appendix 2. Let a = (a1,. .. , a k), b = (b1,. .. , b k), with aj $le$ bj $le$ aj + $pi$, and denote by M a,b the set of families ($alpha$, $beta$) = ($alpha$1, $beta$1),. .. , ($alpha$ k , $beta$ k) such that 1","PeriodicalId":213844,"journal":{"name":"Complex Analysis and Spectral Theory","volume":"276 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124451490","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":"Inner vectors for Toeplitz operators","authors":"R. Cheng, J. Mashreghi, W. Ross","doi":"10.1090/conm/743/14961","DOIUrl":"https://doi.org/10.1090/conm/743/14961","url":null,"abstract":"In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.","PeriodicalId":213844,"journal":{"name":"Complex Analysis and Spectral Theory","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128763935","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":"Boundary values of holomorphic distributions\u0000 in negative Lipschitz classes","authors":"A. O’Farrell","doi":"10.1090/conm/743/14959","DOIUrl":"https://doi.org/10.1090/conm/743/14959","url":null,"abstract":"We consider the behaviour at a boundary point of an open subset $Usubsetmathbb{C}$ of distributions that are holomorphic on $U$ and belong to what are called negative Lipschitz classes. The result explains the significance for holomorphic functions of series of Wiener type involving Hausdorff contents of dimension between $0$ and $1$. We begin with a survey about function spaces and capacities that sets the problem in context and reviews the relevant general theory. The techniques used include the construction of a special partition of the identity that may be of independent interest.","PeriodicalId":213844,"journal":{"name":"Complex Analysis and Spectral Theory","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128334790","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":"A global domination principle for\u0000 𝑃-pluripotential theory","authors":"N. Levenberg, M. Perera","doi":"10.1090/conm/743/14955","DOIUrl":"https://doi.org/10.1090/conm/743/14955","url":null,"abstract":"We prove a global domination principle in the setting of P-pluripotential theory. This has many applications including a general product property for P-extremal functions. The key ingredient is the proof of the existence of a strictly plurisubharmonic P-potential.","PeriodicalId":213844,"journal":{"name":"Complex Analysis and Spectral Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129712941","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":"Jack and Julia","authors":"R. Fournier, Oliver Roth","doi":"10.1090/conm/743/14962","DOIUrl":"https://doi.org/10.1090/conm/743/14962","url":null,"abstract":". We state and prove a multi–point version of Jack’s Lemma for functions not necessarily analytic on the closure of the open unit disc. Our proof in particular does not rely on Julia’s lemma.","PeriodicalId":213844,"journal":{"name":"Complex Analysis and Spectral Theory","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127802515","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}