Hessenberg pairs of linear transformations

Let KK denote a field and VV denote a nonzero finite-dimensional vector space over KK. We consider an ordered pair of linear transformations A:V→VA:V→V and A∗:V→VA∗:V→V that satisfy (i)–(iii) below.1.[(i)]Each of A,A∗A,A∗ is diagonalizable on VV.2.[(ii)]There exists an ordering {Vi}i=0d of the eigenspaces of AA such thatA∗Vi⊆V0+V1+⋯+Vi+1(0⩽i⩽d), where V-1=0V-1=0, Vd+1=0Vd+1=0.3.[(iii)]There exists an ordering {Vi∗}i=0δ of the eigenspaces of A∗A∗ such thatAVi∗⊆V0∗+V1∗+⋯+Vi+1∗(0⩽i⩽δ), where V-1∗=0, Vδ+1∗=0.We call such a pair a Hessenberg pair   on VV. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.