Spectrum and direct integral

Azoff, Edward A.

doi:10.1090/S0002-9947-1974-0350494-4

Spectrum and direct integral
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by Edward A. Azoff PDF

Trans. Amer. Math. Soc. 197 (1974), 211-223 Request permission

Abstract:

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

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