Hessenberg Pairs of Linear Transformations

TL;DR

This paper introduces Hessenberg pairs of linear transformations, characterizes them, and explores their relationship with tridiagonal pairs, expanding the understanding of structured pairs of linear operators.

Contribution

The paper provides new characterizations of Hessenberg pairs and clarifies their connection to tridiagonal pairs, contributing to the theory of structured linear transformations.

Findings

Hessenberg pairs are characterized by specific eigenspace inclusion relations.

Hessenberg pairs are related to but distinct from tridiagonal pairs.

The paper establishes conditions under which Hessenberg pairs can be classified.

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 $A^*: V \to V$ 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 $\lbrace V^*_i \rbrace_{i=0}^{\delta}$ of the eigenspaces of $A^*$ such that A V^*_i \subseteq V^*_0 + V^*_1 + ... +V^*_{i+1} \qquad \qquad (0 \leq i \leq \delta), where $V^*_{-1} = 0$, $V^*_{\delta+1}= 0$. 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.