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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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On analytically invariant subspaces and spectra
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by Domingo A. Herrero PDF
Trans. Amer. Math. Soc. 233 (1977), 37-44 Request permission

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 $\mathcal {A}_T^a$.
References
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • 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