In this paper, all spaces are
separable Hilbert spaces and all operators are bounded linear transformations.
Questions involving the structure of an operator for which an analytic function of it
is normal or which satisfies a polynomial with certain operator coefficients have been
considered and studied separately. Using von Neumann’s reduction theory, a unified
approach to these and similar questions can be given. This method yields
generalizations of the cases which has been previously investigated, including
structure results for n. normal operators. Through reduction theory of von Neumann
algebras, the study of structural questions for a particular orerator is reduced to the
properties of the often simpler, reduced operators. In all of the applications
presented in this paper, the reduced operators will simply involve algebraic
operators.
