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Transactions of the American Mathematical Society

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



On the spectra of the restrictions of an operator

Author: Domingo A. Herrero
Journal: Trans. Amer. Math. Soc. 233 (1977), 45-58
MSC: Primary 47A10; Secondary 47A15
MathSciNet review: 0473870
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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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Keywords: Riesz’ functional calculus, invariant subspace, hyperinvariant subspace, spanning set of subspaces, spectrum, restriction of an operator, spectra of the restrictions, disjointness, component, algebraic sum, algebraic direct sum, operator induced on the quotient space, spectral radius, Hilbert’s theorem on approximation of Jordan curves by lemniscates
Article copyright: © Copyright 1977 American Mathematical Society