Spectral Decompositions Arising from Atzmon’s Hyperinvariant Subspace Theorem

Abstract

By means of a weaker functional model, we prove the existence of non-trivial closed hyperinvariant subspaces for linear bounded operators generalizing, in particular, a classical theorem of Atzmon and revealing the spectral nature of the hyperinvariant subspaces involved. As an application, we show non-trivial spectral subspaces for Bishop operators on $$L^p[0,1)$$ , $$1\le p<\infty $$ , as long as they satisfy Atzmon’s Theorem, providing, in turns, a local spectral decomposition.