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Analytically Decomposable Operators

Ridgley Lange
Transactions of the American Mathematical Society
Vol. 244 (Oct., 1978), pp. 225-240
DOI: 10.2307/1997896
Stable URL: http://www.jstor.org/stable/1997896
Page Count: 16
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Analytically Decomposable Operators
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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 ofanalytically 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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