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.