The structure of some linear transformations

Let F denote an algebraically closed field and let V denote a finite-dimensional vector space over F. Recently Ito and Terwilliger considered a system of linear transformations on V which generalizes the notions of a tridiagonal pair and a q-inverting pair. In their paper they mentioned some open problems about this system. In this paper we solve Problem 1.2 with the following results. Let denote the common eigenspaces of A+,A- and let denote the common eigenspaces of . We show that each of is determined up to affine transformation by the sequences ; . We also show that the following are equivalent: (i) there exists a nonzero bilinear form on V such that 〈A+u,v〉=〈u,A+v〉 and for all u,v∈V; (ii) there exist scalars α,α∗,β,β∗ in F with α,α∗ nonzero such that A-=αA++βI and ; and (iii) both and are tridiagonal pairs.