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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Spectral theory for subnormal operators
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by R. G. Lautzenheiser PDF
Trans. Amer. Math. Soc. 255 (1979), 301-314 Request permission

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 $\sigma (T|\mathcal {M}) \subset K$. 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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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 255 (1979), 301-314
  • MSC: Primary 47B20; Secondary 47A15
  • DOI: https://doi.org/10.1090/S0002-9947-1979-0542882-2
  • MathSciNet review: 542882