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 | References | Similar Articles | Additional Information

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