Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 0350494
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B40

Retrieve articles in all journals with MSC: 47B40

Additional Information

Keywords: Spectrum, direct integral, measurable set-valued function, finite topology, hyperinvariant subspace
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society