Classification of hereditary matrices
Let \({\mathcal H}\) be a complex Hilbert space. For a polynomial \(p\in\mathbb{C}[x, y]\), \(p(x,y)= \sum p_{mn}y^nx^m\), and an operator \(T\in{\mathcal L}({\mathcal H})\) the authors define \(p(T)= \sum p_{mn}T^{*n} T^m\). \(T\) is called a root of \(p\) if \(p(T)= 0\). For the case that \(T\) is an algebraic operator, the authors investigate the relationships which \(p(T)= 0\) forces between \(p\) and \(T\). This relationship is in terms of invariant subspaces of \(T\) and vanishing of coefficients of \(p\), leading to certain graphs. For diagonalizable operators \(T\) and for operators with exactly one or two eigenvalues, the authors characterize \(p(T)= 0\) by graph theoretic conditions.
scientific article; zbMATH DE number 1154659
scientific article