Spectrum and direct integral

Abstract

Let T = ∫ Z ⊕ T ( E ) T = \smallint _Z^ \oplus T(\mathcal {E}) be a direct integral of Hilbert space operators, and equip the collection C \mathcal {C} of compact subsets of C with the Hausdorff metric topology. Consider the [set-valued] function sp which associates with each E ∈ Z \mathcal {E} \in Z the spectrum of T ( E ) T(\mathcal {E}) . The main theorem of this paper states that sp is measurable. The relationship between σ ( T ) \sigma (T) and { σ ( T ( E ) ) } \{ \sigma (T(\mathcal {E}))\} is also examined, and the results applied to the hyperinvariant subspace problem. In particular, it is proved that if σ ( T ( E ) ) \sigma (T(\mathcal {E})) consists entirely of point spectrum for each E ∈ Z \mathcal {E} \in Z , then either T is a scalar multiple of the identity or T has a hyperinvariant subspace; this generalizes a theorem due to T. Hoover.