Some properties of sharp Hessenberg pairs

AbstractLet F denote a field and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable on V; (ii) there exists an ordering {Vi}i=0d of the eigenspaces of A such that A∗Vi⊆V0+V1+⋯+Vi+1 for 0⩽i⩽d, where V-1:=0 and Vd+1:=0; (iii) there exists an ordering {Vi∗}i=0δ of the eigenspaces of A∗ such that AVi∗⊆V0∗+V1∗+⋯+Vi+1∗ for 0⩽i⩽δ, where V-1∗:=0 and Vδ+1∗:=0. We call such a pair a Hessenberg pair on V. It is known that if the Hessenberg pair A,A∗ on V is irreducible then d=δ and for 0⩽i⩽d the dimensions of Vi and Vd-i∗ coincide. We say a Hessenberg pair A,A∗ on V is sharp whenever it is irreducible and dimV0∗=dimVd=1.In this paper, we give the definitions of a Hessenberg system and a sharp Hessenberg system. We discuss the connection between a Hessenberg pair and a Hessenberg system. We also define a finite sequence of scalars called the parameter array for a sharp Hessenberg system, which consists of the eigenvalue sequence, the dual eigenvalue sequence and the split sequence. We calculate the split sequence of a sharp Hessenberg system. We show that a sharp Hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form 〈,〉 on V that satisfies 〈Au,v〉=〈u,Av〉 and 〈A∗u,v〉=〈u,A∗v〉 for all u,v∈V