摘要 : 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 ~(A+), ~(A-),A+,A- on V which generalizes the notions... 展开 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 ~(A+), ~(A-),A+,A- 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 { ~(Vi)}i=0d denote the common eigenspaces of ~(A+), ~(A-) and let {Vi}i=0d denote the common eigenspaces of A+,A-. We show that each of ~(A+), ~(A-),A+,A- is determined up to affine transformation by the sequences { ~(Vi)}i=0d; {Vi}i=0d. 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 〈A+u,v〉=〈u,A+v〉 for all u,v∈V; (ii) there exist scalars α, ~α,β, ~β in F with α, ~α nonzero such that ~(A-)=α ~(A+)+βI and A-= ~αA++ ~βI; and (iii) both ~(A+),A+ and ~(A-),A- are tridiagonal pairs. 收起