线性变换的Hessenberg对
Hessenberg Pairs of Linear Transformations
摘要：
令$\fld$表示域，$V$表示$\fld$上的非零有限维向量空间。我们考虑一对有序线性变换$A: V \to V$和$A^*: V \to V$满足下面的（i）-（iii）。每个$A, A^*$都可以在$V$上对角化。存在$A$的特征空间的序$\lbrace V_i \rbrace_{i=0}^d$，使得A^* V_i \subseteq V_0 + V_1 +. + V_{i+1} \qquad \qquad（0 \leq i \leq d），其中$V_{-1} = 0$，$V_{d+1}= 0$。存在$A^*$的特征空间的序$\lbrace V^*_i \rbrace_{i=0}^δ$，使得A V^*_i \subseteq V^*_0 + V^*_1 +. +V^*_{i+1} \qquad \qquad（0 \leq i \leq δ），其中$V^*_{-1} = 0$，$V^*_{δ+1}= 0$。我们在$V$上称这样的对为{ \it Hessenberg对}。本文给出了Hessenberg对的一些特征。我们还解释了如何海森伯格对有关的三对角对。
英文摘要：
Let $\fld$ denote a field and $V$ denote a nonzero finite-dimensional vector space over $\fld$. We consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to V$ that satisfy (i)--(iii) below. Each of $A, A^*$ is diagonalizable on $V$. There exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that A^* V_i \subseteq V_0 + V_1 + ... + V_{i+1} \qquad \qquad (0 \leq i \leq d), where $V_{-1} = 0$, $V_{d+1}= 0$. There exists an ordering $\lbrace V^*_i \rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that A V^*_i \subseteq V^*_0 + V^*_1 + ... +V^*_{i+1} \qquad \qquad (0 \leq i \leq δ), where $V^*_{-1} = 0$, $V^*_{δ+1}= 0$. We call such a pair a {\it Hessenberg pair} on $V$. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.
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