由线性算子生成的多项式代数
Algebras of Polynomials Generated by Linear Operators
摘要：
设$E$是一个Banach空间，$A$是一个带单位元的交换Banach代数.令${P}(E, A)$是由有界线性算子生成的$E$上的$A$值多项式空间（${P}(E,A)$中的$n$ -齐次多项式的形式为$P=\sum_{i=1}^\infty T^n_i$，其中$T_i:E\to A$（$1\leq i <\infty$）是有界线性算子和$\sum_{i=1}^\infty \|T_i\|^n < \infty$）.对于$E$中的紧集$K$，我们让${P}(K, A)$是${P}(E,A)$中多项式$P$的限制$P|_K$在$C(K,A)$中的闭包。证明了${P}(K, A)$是$A$ -值一致代数，并且在一定条件下，它与单射张量积$\mathcal{P}_N(K)\hat\otimes_εA$等距同构，其中$\mathcal{P}_N(K)$是$K$上由核标值多项式生成的一致代数.然后，将${P}(K, A)$的特征空间标识为$\hat{K}_N\times \mathfrak{M}(A)$，其中$\hat K_N$是$E$中的$K$的核多项式凸包，而$\mathfrak{M}(A)$是$A$的特征空间。
英文摘要：
Let $E$ be a Banach space and $A$ be a commutative Banach algebra with identity. Let ${P}(E, A)$ be the space of $A$-valued polynomials on $E$ generated by bounded linear operators (an $n$-homogenous polynomial in ${P}(E,A)$ is of the form $P=\sum_{i=1}^\infty T^n_i$, where $T_i:E\to A$ ($1\leq i <\infty$) are bounded linear operators and $\sum_{i=1}^\infty \|T_i\|^n < \infty$). For a compact set $K$ in $E$, we let ${P}(K, A)$ be the closure in $C(K,A)$ of the restrictions $P|_K$ of polynomials $P$ in ${P}(E,A)$. It is proved that ${P}(K, A)$ is an $A$-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product $\mathcal{P}_N(K)\hat\otimes_εA$, where $\mathcal{P}_N(K)$ is the uniform algebra on $K$ generated by nuclear scalar-valued polynomials. The character space of ${P}(K, A)$ is then identified with $\hat{K}_N\times \mathfrak{M}(A)$, where $\hat K_N$ is the nuclear polynomially convex hull of $K$ in $E$, and $\mathfrak{M}(A)$ is the character space of $A$.
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