Two commuting operators associated with a tridiagonal pair
- 作者：Sarah Bockting-Conrad bockting@math.wisc.edu
- 关键词：Primary ; 15A21 ; Secondary ; 05E30
- 刊名：Linear Algebra and its Applications
- 出版年：2012
- 期刊代码：38_00243795
- 类别：mt
- 出版时间：1 July, 2012
- 卷：437
- 期：1
- 页码：242-270
- 文件大小：555 K
摘要
Let denote a field and let V denote a vector space over with finite positive dimension. We consider an ordered pair of linear transformations and that satisfy the following four conditions: (i) Each of is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that for , where and ; (iii) there exists an ordering of the eigenspaces of such that for , where and ; (iv) there does not exist a subspace W of V such that , , , . We call such a pair a tridiagonal pair on V. It is known that ; to avoid trivialities assume . We show that there exists a unique linear transformation such that and for . We show that there exists a unique linear transformation such that and for, where and (resp. ) denotes the eigenvalue of A associated with (resp. ). We characterize in several ways. There are two well-known decompositions of V called the first and second split decomposition. We discuss how act on these decompositions. We also show how relate to each other. Along this line we have two main results. Our first main result is that commute. In the literature on TD pairs, there is a scalar used to describe the eigenvalues. Our second main result is that each of is a polynomial of degree d in , under a minor assumption on .