Affine transformations of a sharp tridiagonal pair

作者：Hou Bo; Yang Longmei; Gao Suogang^*

来源：Linear Algebra and Its Applications, 2014, 462: 16-38.

DOI：10.1016/j.laa.2014.08.012

• Tridiagonal pair
• Tridiagonal system
• Affine transformation
• Parameter array
• Sharp

Let K denote a field and let V denote a vector space over K with finite positive dimension. By a tridiagonal pair, we mean an ordered pair A, A* of K-linear transformations from V to V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering {V-i}(i=0)(d) of the eigenspaces of A such that A*V-i subset of Vi-1 + V-i + Vi+1 (0 <= i <= d), where V-1 = 0, Vd+1 = 0; (iii) there exists an ordering {V-i*}(i=0)(delta) of the eigenspaces of A* such that AV(i)* subset of + Vi-1* + V-i* + Vi+1* (0 <= i <= delta), where V-1* = 0, V delta+1* = 0; (iv) there is no subspace W of V such that AW subset of W, A*W subset of W, W not equal 0, W not equal V. It is known that eta A +mu I, eta*A* + mu*I is also a tridiagonal pair on V, where eta,mu,eta*,mu* are scalars in K with eta,eta* nonzero. In this paper we give the necessary and sufficient conditions for these tridiagonal pairs to be isomorphic to A, A* or A*, A. We do this under a mild assumption, called the sharp condition.

• 出版日期2014-12-1
• 单位河北师范大学