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Transactions of the American Mathematical Society

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

 

 

On analytically invariant subspaces and spectra


Author: Domingo A. Herrero
Journal: Trans. Amer. Math. Soc. 233 (1977), 37-44
MSC: Primary 47A15; Secondary 47A10
DOI: https://doi.org/10.1090/S0002-9947-1977-0482289-8
MathSciNet review: 0482289
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Abstract: Let T be a bounded linear operator from a complex Banach space $ \mathfrak{X}$ into itself. Let $ {\mathcal{A}_T}$ and $ \mathcal{A}_T^a$ denote the weak closure of the polynomials and the rational functions (with poles outside the spectrum $ \sigma (T)$ of T) in T, respectively. The lattice $ {\operatorname{Lat}}\;\mathcal{A}_T^a$ of (closed) invariant subspaces of $ \mathcal{A}_T^a$ is a very particular subset of the invariant subspace lattice $ {\operatorname{Lat}}\;{\mathcal{A}_T} = {\operatorname{Lat}}\;T$ of T. It is shown that: (1) If the resolvent set of T has finitely many components, then $ {\operatorname{Lat}}\;\mathcal{A}_T^a$ is a clopen (i.e., closed and open) sublattice of $ {\operatorname{Lat}}\;T$, with respect to the ``gap topology'' between subspaces. (2) If $ {\mathfrak{M}_1},{\mathfrak{M}_2} \in {\operatorname{Lat}}\;T,{\mathfrak{M}_1} \cap {\mathfrak{M}_2} \in {\operatorname{Lat}}\;\mathcal{A}_T^a$ and $ {\mathfrak{M}_1} + {\mathfrak{M}_2}$ is closed in $ \mathfrak{X}$ and belongs to $ {\operatorname{Lat}}\;\mathcal{A}_T^a$, then $ {\mathfrak{M}_1}$ and $ {\mathfrak{M}_2}$ also belong to $ {\operatorname{Lat}}\;\mathcal{A}_T^a$. (3) If $ \mathfrak{M} \in {\operatorname{Lat}}\;T,R$ is the restriction of T to $ \mathfrak{M}$ and $ \bar T$ is the operator induced by T on the quotient space $ \mathfrak{X}/\mathfrak{M}$, then $ \sigma (T) \subset \sigma (R) \cup \sigma (\bar T)$. Moreover, $ \sigma (T) = \sigma (R) \cup \sigma (\bar T)$ if and only if $ \mathfrak{M} \in {\operatorname{Lat}}\;\mathcal{A}_T^a$. The results also include an analysis of the semi-Fredholm index of R and $ \bar T$ at a point $ \lambda \in \sigma (R) \cup \sigma (\bar T)\backslash \sigma (T)$ and extensions of the results to algebras between $ {\mathcal{A}_T}$ and $ \mathcal{A}_T^a$.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0482289-8
Keywords: Invariant subspace, analytically invariant subspace, bi-invariant subspace, lattice, spectrum, gap topology, restriction of an operator, operator induced on the quotient space, semi-Fredholm operator, index
Article copyright: © Copyright 1977 American Mathematical Society