Algebraic properties of operator roots of polynomials

Properties of m-selfadjoint and m  -isometric operators have been investigated by several researchers. Particularly interesting to us are algebraic properties of nilpotent perturbations of such operators. McCullough and Rodman showed in the nineties that if Qn=0Qn=0 and A is a selfadjoint operator commuting with Q   then the sum A+QA+Q is a (2n−1)(2n−1)-selfadjoint operator. Very recently, Bermúdez, Martinón, and Noda proved a similar result for nilpotent perturbations of isometries. Via a new approach, we obtain simple proofs of these results and other generalizations to operator roots of polynomials.