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

✍ Scribed by Carl de Boor; Amos Ron; Zuowei Shen

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 865 KB
• Volume: 123
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1996.0072

✦ Synopsis

Given an index set X, a collection B of subsets of X, and a collection (l x : x # X) of commuting linear maps on some linear space, the family of linear operators whose joint kernel K=K(B) is sought consists of all l A :=> a # A l a with A any subset of X which intersects every B # B. It is shown that certain conditions on B and l, used in a previous paper to obtain the inequality dim K(B) :

or the corresponding equality, can be weakened. For example, the additional assumption of equicardinality of the elements of B can be dropped. However, the notion of ``placeability'' continues to play an essential role. The results are then described in the rather different language employed by W. Dahmen, A. Dress, and C. A. Micchelli (Adv. in Appl. Math. 17 (1996), 251 307) to facilitate comparisons.