Hessenberg Pairs of Linear Transformations

Abstract

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 $A^{\ast }:V\to V$ that satisfy (i)--(iii) below. Each of $A,A^{\ast }$ is diagonalizable on $V$. There exists an ordering $\{V_i{\}}_{i=0}^d$ 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 $\{V_i^{\ast }{\}}_{i=0}^{\delta }$ of the eigenspaces of $A^{\ast }$ such that A V^*_i \subseteq V^*_0 + V^*_1 + ... +V^*_{i+1} \qquad \qquad (0 \leq i \leq δ), where $V_{-1}^{\ast }=0$, $V_{\delta +1}^{\ast }=0$. 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.

10 pages

Cited by in corpus (1)

• On the shape of a tridiagonal pair