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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spectrum and direct integral


Author: Edward A. Azoff
Journal: Trans. Amer. Math. Soc. 197 (1974), 211-223
MSC: Primary 47B40
DOI: https://doi.org/10.1090/S0002-9947-1974-0350494-4
MathSciNet review: 0350494
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0350494-4
Keywords: Spectrum, direct integral, measurable set-valued function, finite topology, hyperinvariant subspace
Article copyright: © Copyright 1974 American Mathematical Society