Complex Analysis and Spectral Theory最新文献

A panorama of positivity. II: Fixed dimension 积极的全景。二、固定尺寸

Complex Analysis and Spectral Theory

Pub Date : 2020-03-31 DOI: 10.1090/conm/743/14958

A. Belton, D. Guillot, A. Khare, M. Putinar

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.

本调查包含由正半确定性(矩阵或核)概念统一的主题选择，反映了对离散数据(图或网络)或连续对象(概率或质量分布)施加的自然约束。我们强调以各种形式保持积极性的入门式操作。从调和分析，函数理论，算子理论，统计学，组合学和群表示的技术被调用。一些部分被遗忘的经典根在度量几何和距离变换提出了评论和完整的参考书目。包括高维协方差估计和正则化的现代应用。

引用次数: 7

A holomorphic functional calculus for finite families of commuting semigroups 交换半群有限族的全纯泛函演算

Complex Analysis and Spectral Theory

Pub Date : 2019-01-02 DOI: 10.1090/conm/743/14956

J. Esterle

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

设A是一个交换巴拿赫代数，使得uA = {0} 为你 $in$ a {0} 它拥有丰富的主要理想。本文的目的是给出一个定义F(——)的一般框架$lambda$1$Delta$1、……， --$lambda$ k $Delta$T k)，其中F属于定义在C k的合适开子集上的全纯函数的自然类，该类包含(——)的“Arveson谱”$lambda$1$Delta$1、……， --$lambda$ k $Delta$T (k)，其中 $Delta$1、……， $Delta$T k是A上的乘子交换单参数半群的无限小发生器，这些半群属于下列类之一(1)强连续半群T = (T (ia) T >0，使得 $cup$t>0T (ia)A在A中密度大，其中A $in$ R.(2)半群的类T = (T)$zeta$）) $zeta$$in$sa,b在开扇区sa,b上是全纯的，使得T ($zeta$)A在A中对某些是稠密的，或者对所有都是等价的 $zeta$ $in$ S a b。我们使用了拟乘子的概念，这是作者在1981年的Banach代数长滩会议上提出的:所考虑的半群的生成将被定义为A上的拟乘子，对于 $zeta$ 在Arveson解决方案集中 $sigma$ar()$Delta$T)解决方案($Delta$—— $zeta$I)—1将被定义为a上的正则拟乘子，即a上的拟乘子S使其大于n$ge$1 $lambda$ n nsu 0和某个u生成a的稠密理想并且属于nsn, n的定义域的交 $ge$ 1. 第一步是“规范化”巴拿赫代数A，即连续地将A嵌入到一个巴拿赫代数B中，B与A具有相同的准乘子代数，但它是线性的$rightarrow$0 + T (ia) M(B) < +$infty$ 如果T属于类(1)，并且对于该类(1 $zeta$$rightarrow$0 $zeta$$in$s $alpha$，$beta$ T ($zeta$) < +$infty$ 对于所有配对($alpha$， $beta$)使得a < $alpha$ < $beta$ < b，如果T属于类(2)。迭代这个过程允许考虑($lambda$j$Delta$tj + $zeta$I) -1作为M(B)的一个元素 $zeta$ $in$ 研究人员(——$lambda$j$Delta$T j)表示——的“Arveson解析集”$lambda$j$Delta$tj，并且即使给定的半群在原点附近没有界，也可以使用标准的积分“解公式”。函数演算的第一种方法涉及到在附录2中描述的速降函数代数的对偶G A,b。令a = (a1，.…)， a k)， b = (b1，…， b k)，与aj $le$ bj $le$ Aj + $pi$，用M a,b表示($alpha$， $beta$) = ($alpha$1， $beta$1)……， ($alpha$ K， $beta$ K)使

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引用次数: 0

Inner vectors for Toeplitz operators Toeplitz算子的内向量

Complex Analysis and Spectral Theory

Pub Date : 2018-09-11 DOI: 10.1090/conm/743/14961

R. Cheng, J. Mashreghi, W. Ross

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.

本文综述并汇集了几种获取Toeplitz算子内函数的方法。这些方法包括经典定义、Wold分解、算子值泊松积分和Clark测度。然后我们将这些概念扩展到模型空间上的内部函数。在此过程中，我们提出了一些新颖的例子。

引用次数: 2

Boundary values of holomorphic distributions in negative Lipschitz classes 负Lipschitz类全纯分布的边值

Complex Analysis and Spectral Theory

Pub Date : 2018-06-26 DOI: 10.1090/conm/743/14959

A. O’Farrell

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.

我们考虑一个开放子集$U子集mathbb{C}$的边界点上的行为，这些分布在$U$上是全纯的，属于所谓的负Lipschitz类。这一结果解释了维数在$0$和$1$之间的Hausdorff内容的Wiener型级数全纯函数的意义。我们首先对功能空间和能力进行调查，将问题置于上下文中，并回顾相关的一般理论。所使用的技术包括构建可能具有独立兴趣的身份的特殊划分。

引用次数: 2

A global domination principle for 𝑃-pluripotential theory 𝑃-pluripotential理论的全局支配原则

Complex Analysis and Spectral Theory

Pub Date : 2018-06-17 DOI: 10.1090/conm/743/14955

N. Levenberg, M. Perera

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.

在p -多势理论的背景下，我们证明了一个全局支配原则。这有许多应用，包括p极值函数的一般乘积性质。关键是证明了严格的多次谐波p势的存在。

引用次数: 6

Jack and Julia 杰克和朱莉娅

Complex Analysis and Spectral Theory

Pub Date : 1900-01-01 DOI: 10.1090/conm/743/14962

R. Fournier, Oliver Roth

. 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.

。我们陈述并证明了开单位盘闭包上不一定解析函数的杰克引理的多点版本。我们的证明尤其不依赖于茱莉亚引理。

引用次数: 0