\(m\)-isometries, \(n\)-symmetries and other linear transformations which are hereditary roots (Q1946564)

The author develops techniques to study bounded linear transformations \(T\) of a complex Hilbert space such that \(\sum_{m,n} c_{m,n} T^{*n} T^m=0\), where \(c_{m,n} \in C\) and all but finitely many \(c_{m,n}\) are zero. Such an operator is called a hereditary root of the polynomial \(p(x,y)=\sum_{m,n} c_{m,n} y^{n} x^m\). The techniques developed include finding maximal invariant subspaces satisfying certain operator-theoretic properties, a description of the spectral picture of a root, resolvent inequalities for a root, and additional properties which can be derived when knowing more about the spectrum of a particular root.