R. Killip, C. Remling
Reducing Subspaces
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ABSTRACT. Let $T$ be a self-adjoint operator acting in a separable Hilbert space.
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 the Hilbert space 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.