设A是一个交换Banach代数，使得对于u $\in$ A \{0}，u A = {0}且u A具有稠密主理想.给出了定义F（-- $λ$ 1 $Δ$ T1，. ..，- $λ$ kPDMATHABMO T k）其中F属于定义在C k的适当开子集上的全纯函数的自然类，所述开子集包含（- $λ$ 1 $Δ$ T 1，. .. .，- $λ$ k $Δ$ T k），其中$Δ$ T 1，. ..，$Δ$ T k是A上的可交换单参数乘子半群的无穷小生成元，它们属于下列一类：（1）强连续半群T =（T（te ia）t>0，使得$\cup$ t> 0 T（te ia）A在A中稠密，其中一个$\in$ R. (2)半群类T =（T（$ζ$））$ζ$ $\in$ Sa，b在开扇区Sa，b上全纯，使得T（$ζ$）A在A中对某些或等价地对所有$ζ$ $\in$ Sa，b稠密.我们使用的概念quasimulator，介绍了作者在1981年长滩会议上的Banach代数：所考虑的半群的生成元将被定义为A上的拟乘子，并且对于Arveson预解集$σ$中的$ζ$，（$Δ$ T）解决方案（$Δ$ T -- $ζ$ I）将被定义为A上的规则准乘数，即A上的一个拟乘数S使得对于某些$λ$ > 0和某些生成A的稠密理想且属于Sn，n $\ge$ 1的域的交集的u，supn $\ge$ 1 $λ$ n S n u < + $\infty$。第一步在于“规范化”Banach代数A，即，将A连续嵌入到具有与A相同的准乘子代数的Banach代数B中，但是对于该Banach代数B，如果T属于类（1），则lim sup $\rightarrow$ 0 + T（te ia）M（B）< + $\infty$，并且对于该Banach代数B，lim sup $ζ$ $\rightarrow$ 0 $ζ$ $\in$ S $α$，对于所有对（$α$，$β$），$β$ T（$ζ$）< + $\infty$，使得如果T属于类（2），则a < $α$ < $β$ < b。迭代此过程，这允许考虑（$λ$ j $Δ$ T j + $ζ$ I）-1作为$ζ$ $\in$ Resar（-- $λ$ j $Δ$ T j）的M（B）的元素，即-- $λ$ j $Δ$ T j的“Arveson预解式集“，并且即使给定半群在原点附近没有边界，也可以使用标准积分”预解式公式“。谓词演算的第一种方法涉及到快速下降函数代数的对偶G a，b，在附录2中描述。设a =（a1，. ..，ak），b =（b1，. ..，b，k），其中aj $\le$ bj $\le$ aj + $π$，并且由Ma，b表示族的集合（$α$，$β$）=（$α$ 1，$β$ 1），. ..，（$α$ k，$β$ k），使得1
英文摘要：
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 (--$λ$1$Δ$T 1 ,. .. , --$λ$ k $Δ$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 (--$λ$1$Δ$T 1 ,. .. , --$λ$ k $Δ$T k), where $Δ$T 1 ,. .. , $Δ$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 ($ζ$)) $ζ$$\in$S a,b holomorphic on an open sector S a,b such that T ($ζ$)A is dense in A for some, or equivalently for all $ζ$ $\in$ S a,b. We use the notion of quasimultiplier, introduced in 1981 by the author at the Long Beach Conference on Banach algebras: the generators of the semigroups under consideration will be defined as quasimultipliers on A, and for $ζ$ in the Arveson resolvent set $σ$ar($Δ$T) the resolvent ($Δ$T -- $ζ$I) --1 will be defined as a regular quasimultiplier on A, i.e. a quasimultiplier S on A such that sup n$\ge$1 $λ$ n S n u < +$\infty$ for some $λ$ > 0 and some u generating a dense ideal of A and belonging to the intersection of the domains of S n , n $\ge$ 1. The first step consists in "normalizing" the Banach algebra A, i.e. continuously embedding A in a Banach algebra B having the same quasi-multiplier algebra as A but for which lim sup t$\rightarrow$0 + T (te ia) M(B) < +$\infty$ if T belongs to the class (1), and for which lim sup $ζ$$\rightarrow$0 $ζ$$\in$S $α$,$β$ T ($ζ$) < +$\infty$ for all pairs ($α$, $β$) such that a < $α$ < $β$ < b if T belongs to the class (2). Iterating this procedure this allows to consider ($λ$j$Δ$T j + $ζ$I) --1 as an element of M(B) for $ζ$ $\in$ Resar(--$λ$j$Δ$T j), the "Arveson resolvent set " of --$λ$j$Δ$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 + $π$, and denote by M a,b the set of families ($α$, $β$) = ($α$1, $β$1),. .. , ($α$ k , $β$ k) such that 1
Indian Institute of Science（印度科学研究所）；Jawaharlal Nehru Centre for Advanced Scientific Research（贾瓦哈拉尔·尼赫鲁高级科学研究中心）；SRM Institute of Science and Technology（SRM科学与技术学院）
$H^\infty$ Functional Calculus for a Commuting Pair of $\text{Ritt}_{\text{E}}$ Operators
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
