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Transactions of the American Mathematical Society

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

 

 

Almost isolated spectral parts and invariant subspaces


Author: C. R. Putnam
Journal: Trans. Amer. Math. Soc. 216 (1976), 267-277
MSC: Primary 47A10; Secondary 47B20
MathSciNet review: 0385599
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Abstract: Let T be an operator with spectrum $ \sigma (T)$ on a Hilbert space. A compact subset E of $ \sigma (T)$ is called a disconnected part of $ \sigma (T)$ if, for some bounded open set A, E is the closure of $ \sigma (T) \cap A$ and $ \sigma (T) - E$ is the union of the isolated parts of $ \sigma (T)$ lying completely outside the closure of A. The set E is called an almost isolated part of $ \sigma (T)$ if, in addition, the set A can be chosen so as to have a rectifiable boundary $ \partial A$ on which the subset $ \sigma (T) \cap \partial A$ has arc length measure 0. The following results are obtained. If T is subnormal and if E is a disconnected part of $ \sigma (T)$ then there exists a reducing subspace $ \mathfrak{M}$ of T for which $ \sigma (T\vert\mathfrak{M}) = E$. If $ {T^\ast}$ is hyponormal and if E is an almost isolated part of $ \sigma (T)$ then there exists an invariant subspace $ \mathfrak{M}$ of T for which $ \sigma (T\vert\mathfrak{M}) = E$. An example is given showing that if T is arbitrary then an almost isolated part of $ \sigma (T)$ need not be the spectrum of the restriction of T to any invariant subspace.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0385599-7
Keywords: Isolated part of spectrum, disconnected part of spectrum, almost isolated part of spectrum, invariant subspaces, subnormal operators, hyponormal operators
Article copyright: © Copyright 1976 American Mathematical Society