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. If H 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.