Hessenberg Pairs of Linear Transformations

Let $\text{fld}$ denote a field and $V$ denote a nonzero finite-dimensional vector space over $\text{fld}$. We consider an ordered pair of linear transformations $A: V \to V$and$A^: V \to V$thatsatisfy(i)--(iii)below\mathrm{.}Eachof$A, A^$ 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 \delta), 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.