超不变、特征子空间与标记子空间
Hyperinvariant, characteristic and marked subspaces
摘要：
设$V$是域$K$上的有限维向量空间，$f$是$V$的$K$-自同态.在本文中，我们研究了三种类型的$f$不变子空间，即超不变子空间，这是不变的所有自同态的$V$与$f$交换，特征子空间，这是保持固定的所有自同构的$V$与$f$交换，和标记子空间，其中有一个约旦基（关于$f_{|X}$），可以扩展到约旦基的$V$。我们证明了一个子空间是超不变的当且仅当它是特征的和标记的。如果$K$有两个以上的元素，那么每个特征子空间是超不变的。
英文摘要：
Let $V$ be a finite dimensional vector space over a field $K$ and $f$ a $K$-endomorphism of $V$. In this paper we study three types of $f$-invariant subspaces, namely hyperinvariant subspaces, which are invariant under all endomorphisms of $V$ that commute with $f$, characteristic subspaces, which remain fixed under all automorphisms of $V$ that commute with $f$, and marked subspaces, which have a Jordan basis (with respect to $f_{|X}$) that can be extended to a Jordan basis of $V$. We show that a subspace is hyperinvariant if and only if it is characteristic and marked. If $K$ has more than two elements then each characteristic subspace is hyperinvariant.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
