On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators. II (Q1352288)

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 \(B\in\mathbb{B}\).