M-Isometries, n-Symmetries and Other Linear Transformations Which are Hereditary Roots

Abstract

We develop techniques to study bounded linear transformations T such that \({\sum_{m,n} c_{m,n} T^{*n}T^m = 0}\) where \({c_{m,n} \in {\bf C}}\) and all but finitely many cm,n are zero. Such an operator is called a hereditary root of the polynomial \({p(x,y) = \sum_{m,n} c_{m,n} y^{n}x^{m}}\) . Self-adjoint operators, isometries, n-symmetric operators and m-isometries are all examples of roots of polynomials. The techniques developed include finding maximal invariant subspaces satisfying certain operator-theoretic properties, descriptions 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. The results involving maximal invariant subspaces are applicable to operators which are not roots of polynomials.