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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analytically decomposable operators


Author: Ridgley Lange
Journal: Trans. Amer. Math. Soc. 244 (1978), 225-240
MSC: Primary 47B40
MathSciNet review: 506617
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Abstract: The author introduces the notion of an analytically decomposable operator which generalizes the decomposable operator due to C. Foias in that the spectral decompositions of the underlying Banach space (1) admit a wider class of invariant subspaces called ``analytically invariant'' and (2) span the space only densely. It is shown that analytic decomposability is stable under the functional calculus, direct sums and restrictions to certain kinds of invariant subspaces, as well as perturbation by commuting scalar operators. It is fundamental for many of these results that every analytically decomposable operator has the single-valued extension property. An extensive investigation of analytically invariant subspaces is given. The author shows by example that this class is distinct from those of spectral maximal and hyperinvariant subspaces, but he further shows that analytically invariant subspaces have many useful spectral properties. Some applications of the general theory are made. For example, it is shown that under certain restrictions an analytically decomposable operator is decomposable.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0506617-0
PII: S 0002-9947(1978)0506617-0
Keywords: Analytically invariant subspace, analytically decomposable operator, decomposable operator, hyperinvariant, single-valued extension property, spectral maximal
Article copyright: © Copyright 1978 American Mathematical Society