On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators. II
type: Item
instance of: scholarly article
title: On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators. II
zbMATH Open document ID: 0917.47001
author: A. Ron
author: Zuowei Shen
author: Carl De Boor
published in: Advances in Mathematics
publication date: 1998-05-10T00:00:00Z
review text: Let \(X\) be an index set. Let \(\mathbb{B}\subset 2^X\) be such that \(A\subset B\) implies \(A=B\) for any \(A,B\in\mathbb{B}\). Let \(\{\ell_x: x\in X\}\) be a family of commuting linear operators on a linear space. Its joint kernel is \[ k(\mathbb{B})= \Biggl\{\ell_A= \prod_{a\in A}\ell_a: A\in X,\;B\in\mathbb{B},\;A\cap B\neq\emptyset\Biggr\}. \] It is shown that conditions imposed on \(\mathbb{B}\) and \(\ell_a\) in the authors' paper [Part I, Adv. Appl. Math. 17, No. 3, 209-250 (1996)] in order to obtain the inequality/equality \[ \dim k(\mathbb{B})\leq \sum_{B\in\mathbb{B}}\dim k(\{B\}) \] can be weakened. For instance, elements of \(\mathbb{B}\) do not need to have the same cardinality. On the other hand, the assumed placeability is still essential. A set \(Y\) is said to be placeable into \(B\) if \(Y\cup C\in\mathbb{B}\) for a \(C\subseteq\mathbb{B}\) and \(\mathbb{B}\)-placeable if it is placeable for every \(B\in\mathbb{B}\).
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 977960
zbMATH Keywords: commuting linear operators
zbMATH Keywords: equicardinality
zbMATH Keywords: tree-condition
zbMATH Keywords: joint kernel
zbMATH Keywords: placeable
reviewed by: Danuta Przeworska-Rolewicz
MaRDI profile type: Publication
full work available at URL: aima.1996.0072
OpenAlex ID: W2031095402
DOI: 10.1006/AIMA.1996.0072
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description: scientific article; zbMATH DE number 977960