On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators, II

Carl de Boor, Amos Ron, Zuowei Shen
1996 Advances in Mathematics  
Given an index set X, a collection IB of subsets of X, and a collection ('x : x 2 X) of commuting linear maps on some linear space, the family of linear operators whose joint kernel K = K(IB) is sought consists of all'A := Q a2A'a with A any subset of X which intersects every B 2 IB. It is shown that certain conditions on IB and', used in BRS] to obtain the inequality or the corresponding equality, can be weakened. For example, the additional assumption of equicardinality of the elements of IB
more » ... sed there is dropped. However, the notion of'placeability' introduced in BRS] continues to play an essential role. These results are then described in the rather di erent language employed in (the nal version of) DDM] to facilitate comparisons. We note that, in contrast to BRS], we do not assume in this de nition that #Y = #(BnC). E.g., with IB = ffag; fb; cgg, a is IB-placeable. De nition 2.2. A IB-tree is any binary tree with the following properties: (i) The nodes are of the form IB jY nZ for certain Y , Z X with Y \ Z = ;. (ii) Each node is either a leaf, in which case it contains fewer than 2 elements, or else, it is the disjoint union of its two children, IB jY bnZ and IB jY nZ b (the latter may possibly be empty), for some b 2 Xn(Y Z). (iii) IB is the root of this tree. The nonempty leaves of a IB-tree constitute the partition of IB into its elements. De nition 2.3. A IB-tree is placeable if, for each of its nodes, the element used to split that node is placeable in that node. We say that IB satis es the tree-condition if there is a placeable IB-tree. Since the tree-condition involves placeability within each node, any branch of a placeable tree is itself placeable. Hence, any node of a placeable IB-tree satis es itself the tree-condition. Conversely, if b is placeable, and both IB jb and IB nb satisfy the tree-condition, then so does IB. A placeable IB-tree is what is called an IE-tree in BRS] except for the more general de nition 2.1 of placeability used here, and for the fact that empty nodes are allowed here. The latter is a convenience in certain proofs. However, we also have the following.
doi:10.1006/aima.1996.0072 fatcat:2t2cbmgyubey3kpnj4w4gjx3ru