Reducing subspaces

Let T be a self-adjoint operator acting in a separable Hilbert space H. We establish a correspondence between the reducing subspaces of T that come from a spectral projection and the convex, norm-closed bands in the set of finite Borel measures on R. IfH is not separable, we still obtain a reducing subspace corresponding to each convex norm-closed band. These observations lead to a unified treatment of various reducing subspaces; moreover, they also settle some open questions and suggest new decompositions. 1 Reducing subspaces and bands Throughout this paper, we fix a self-adjoint operator T acting in Hilbert space H. As T is self-adjoint, it admits the representation T = ∫ R λ dE(λ) where E(·) is a projection-operator-valued measure. Also, to each ψ ∈ H, we associate its spectral measure, ρψ(M) = ‖E(M)ψ‖.