论文标题
在Banach空间上分离多项式的群体不变
Group invariant separating polynomials on a Banach space
论文作者
论文摘要
我们在Banach Space $ x $上研究了组不变的连续多项式，该$ x $将给定的$ k $ in $ x $和$ k $之外的点$ z $。 We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomial $Q$, then $z$ can also be separated from $K$ by a $G$-invariant continuous多项式$ P $。事实证明，当$ x $是一个复杂的Banach空间时，该结果不得不成立，因此我们提出了一些其他条件，以获得复杂情况的类似结果。我们还假设$ x $具有Schauder基础，该基础为几个古典组提供了申请，我们还获得了分离定理。在这种情况下，我们获得了可以通过封闭的单位球与组不变多项式分开的点的特征。
We study the group invariant continuous polynomials on a Banach space $X$ that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomial $Q$, then $z$ can also be separated from $K$ by a $G$-invariant continuous polynomial $P$. It turns out that this result does not hold when $X$ is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that $X$ has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.
