A general framework for a multi-operator functional calculus
A general framework for a multi-operator functional calculus
复制标题
多算子泛函计算的通用框架
DOI：
10.1016/0001-8708(72)90017-5
复制
发表时间：
1972
影响因子：
1.7
通讯作者：
Joseph L. Taylor
中科院分区：
文献类型：
--
作者：
Joseph L. Taylor
Banach空间X上算子a的解析泛函演算可以这样考虑：如果P是单变量复数的代数，则算子a决定作用（P， X）+ P (u) X：P xx-+ X （P在X上）也就是X上的P模结构。现在代数P通常嵌入到每个拓扑代数%(U)中，其中c @是一个定义域，$ I (U)是U上全纯的函数代数。的确，P对X的作用（由a决定）扩展到U (U)对X的连续作用。这个问题的解是众所周知的，并且是许多现代分析中的关键工具：作用扩展当且仅当U包含算子a的谱。解析泛函微积分也可以看作是关于可交换巴拿赫代数元素的结果。在这种形式中，有一个多变量的版本，即希洛夫-阿伦斯-卡尔德隆定理（参见[1,13]），它指出，如果a，，…， a，是交换巴拿赫代数a的元素，则代数同态p+ p （l，…）， a,): P，+ a扩展到连续同态f+ f （a,，…）， a,): 2I (U)+ a，当U是包含n元组（ur，…）的联合谱的fZn中的定义域时,,)。我们在[14]中引入了Banach空间上算子的交换n元组的联合谱的概念，并在[15]中证明了相应版本的Shilov-Arens-Calderon定理：， a,)决定一个动作(p, 3)+~（a,，…）， a&: P， x x—+ x在x上的n变量多项式代数；这一行动
The analytic functional calculus for an operator a on a Banach space X can be thought of in the following way: If P is the algebra of complex polynomials in one variable, then the operator a determines an action (p, x)+ p (u) x: P xx-+ X of P on X-that is, a P-module structure on X. Now the algebra P is canonically embedded in each of the topological algebras ‘%(U), where UC@ is a domain and ‘$ I (U) is the algebra of functions holomorphic on U. The analytic functional calculus problem for a is the problem of deciding for which domains U, it is true that the action of P on X (determined by a) extends to a continuous action of ‘U (U) on X. The solution to this problem is well known and is a key tool in much of modern analysis: The action extends if and only if U contains the spectrum of the operator a. The analytic functional calculus can also be regarded as a result concerning elements of a commutative Banach algehra. In this form, there is a several variable version, the Shilov-Arens-Calderon Theorem (cf.[I, 13]), which states that if a,,..., a, are elements of a commutative Banach algebra A, then the algebra homomorphism p+ p (ul,..., a,): P,+ A extends to a continuous homomorphism f+ f (a,,..., a,): 2I (U)+ A whenever U is a domain in fZn which contains the joint spectrum of the n-tuple (ur,..., a,). We introduced in [14] a notion of joint spectrum for a commuting n-tuple of operators on a Banach space and in [15] proved the corresponding version of the Shilov-Arens-Calderon Theorem: Such an n-tuple (ui,..., a,) determines an action (p, 3)+~(a,,..., a&: P, x X--+ X of the n-variable polynomial algebra on X; this action