Supernormal mappings: Part 1. Orthogonally indecomposable modules and applications

Let V be a finite-dimensional vector space over a field and f:V×V→K a regular -hermitian form. A linear mapping π:V→V with adjoint mapping π∗, i.e. f(vπ,w)=f(v,wπ∗) for all v,w∈V, is called supernormal or polynomially normal or simply s-normal if s(π)=π∗ holds true for some polynomial s∈K[x]. If π is a unitary transformation or if π is self-adjoint or if π is anti-self-adjoint then π is s-normal. For s-normal mappings π a classification of orthogonally indecomposable π-modules is obtained. The classification distinguishes four types and depends on the form f, the minimum polynomial of π and whether π∗-π is nilpotent or not. We prove a uniqueness statement for orthogonal decompositions (into orthogonally indecomposable modules) of similar s-normal mappings. As an application we generalize the fact that in an orthogonal group each element is a product of two involutions. In the generalized setting both factors are self-adjoint or anti-self-adjoint and the first one is an involution. The last section establishes a decomposition of V into orthogonally indecomposable π-modules such that each π-module is also invariant under both given factors. Part 2 will be devoted to isometric similarity of s-normal mappings.