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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
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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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