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 $L^p[0,1)$, $1 \leq p < \infty$, as long as they satisfy Atzmon’s Theorem, providing, in turns, a local spectral decomposition.