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Transactions of the American Mathematical Society

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



Spectral theory for subnormal operators

Author: R. G. Lautzenheiser
Journal: Trans. Amer. Math. Soc. 255 (1979), 301-314
MSC: Primary 47B20; Secondary 47A15
MathSciNet review: 542882
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Abstract: We give an example of a subnormal operator T such that $ {\text{C}}\,\backslash \,\sigma (T)$ has an infinite number of components, $ \operatorname{int} (\sigma (T))$ has two components U and V, and T cannot be decomposed with respect to U and V. That is, it is impossible to write $ T\, = \,{T_1}\, \oplus \,{T_2}$ with $ \sigma ({T_1})\, = \,\overline U $ and $ \sigma ({T_2})\, = \,\overline V $. This example shows that Sarason's decomposition theorem cannot be extended to the infinitely-connected case.

We also use Mlak's generalization of Sarason's theorem to prove theorems on the existence of reducing subspaces. For example, if X is a spectral set for T and $ K\, \subset \,X$, conditions are given which imply that T has a nontrivial reducing subspace $ \mathcal{M}$ such that $ \sigma (T\vert\mathcal{M})\, \subset \,K$. In particular, we show that if T is a subnormal operator and if $ \Gamma $ is a piecewise $ {C^2}$ Jordan closed curve which intersects $ \sigma (T)$ in a set of measure zero on $ \Gamma $, then $ T\, = \,{T_1}\, \oplus \,{T_2}$ with $ \sigma ({T_1})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname{ext} (\Gamma )} $ and $ \sigma ({T_2})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname{int} (\Gamma )} $.

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Keywords: Spectral set, subnormal operator, Gleason part, peak set
Article copyright: © Copyright 1979 American Mathematical Society