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 into itself. Let and denote the weak closure of the polynomials and the rational functions (with poles outside the spectrum of *T*) in *T*, respectively. The lattice of (closed) invariant subspaces of is a very particular subset of the invariant subspace lattice of *T*. It is shown that: (1) If the resolvent set of *T* has finitely many components, then is a clopen (i.e., closed and open) sublattice of , with respect to the ``gap topology'' between subspaces. (2) If and is closed in and belongs to , then and also belong to . (3) If is the restriction of *T* to and is the operator induced by *T* on the quotient space , then . Moreover, if and only if . The results also include an analysis of the semi-Fredholm index of *R* and at a point and extensions of the results to algebras between and .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0482289-8

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