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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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On the spectra of the restrictions of an operator
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by Domingo A. Herrero PDF
Trans. Amer. Math. Soc. 233 (1977), 45-58 Request permission

Abstract:

Let T be a bounded linear operator from a complex Banach space $\mathfrak {X}$ into itself and let $\mathfrak {M}$ be a closed invariant subspace of T. Let $T|\mathfrak {M}$ denote the restriction of T to $\mathfrak {M}$ and let $\sigma$ denote the spectrum of an operator. The main results say that: (1) If $\mathfrak {X}$ is the closed linear span of a family $\{ {\mathfrak {M}_v}\}$ of invariant subspaces, then every component of $\sigma (T)$ intersects the closure of the set ${ \cup _v}\sigma (T|{\mathfrak {M}_v})$ and every point of $\sigma (T)\backslash { \cup _v}\sigma (T|{\mathfrak {M}_v})$ is an approximate eigenvalue of T. (2) If $\mathfrak {X}$ is the closed linear span of a finite family $\{ {\mathfrak {M}_1}, \ldots ,{\mathfrak {M}_n}\}$ of invariant subspaces, and the spectra $\sigma (T|{\mathfrak {M}_j}),j = 1,2, \ldots ,n$, are pairwise disjoint, then $\mathfrak {X}$ is actually equal to the algebraic direct sum of the ${\mathfrak {M}_j}$’s, the ${\mathfrak {M}_j}$’s are hyperinvariant subspaces of T and $\sigma (T) = \cup _{j = 1}^n\sigma (T|{\mathfrak {M}_j})$. This last result is sharp in a certain specified sense. The results of (1) have a “dual version” $(1’)$; (1) and $(1’)$ are applied to analyze the spectrum of an operator having a chain of invariant subspaces which is “piecewise well-ordered by inclusion", extending in several ways recent results of J. D. Stafney on the spectra of lower triangular matrices.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 233 (1977), 45-58
  • MSC: Primary 47A10; Secondary 47A15
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0473870-0
  • MathSciNet review: 0473870