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Transactions of the American Mathematical Society

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



Analytically invariant and bi-invariant subspaces

Authors: Domingo Antonio Herrero and Norberto Salinas
Journal: Trans. Amer. Math. Soc. 173 (1972), 117-136
MSC: Primary 47A15
MathSciNet review: 0312294
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Abstract: The purpose of this paper is to call attention to some interesting weakly closed algebras related to a bounded linear operator $T$ acting on a Banach space $\mathfrak {X}$ and their associated lattices of invariant subspaces, namely, the algebras generated by the polynomials and by the rational functions in $T$, and the commutant and the double-commutant of $T$. The relationship between those algebras and their lattices, as well as the ones corresponding to the operators induced by $T$ on an invariant subspace (restriction), or on the quotient space $\mathfrak {X}/\mathfrak {M}$ (where $\mathfrak {M}$ is an invariant subspace of a given type) are analyzed. Several results relative to the decomposition of invariant subspaces and the topological structure of the lattices (under the “gap-between-subspaces” metric topology) are also considered.

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Keywords: Lattices of invariant, analytically invariant, bi-invariant, hyperinvariant subspaces, complementary invariant subspaces, decomposition, polynomially generated algebra, analytically generated algebra, commutant, double commutant, reflexive algebra, relatively reflexive algebra, splitting algebra, splitting lattice, gap-metric topology for invariant subspaces, restriction, quotient, normal operators, subnormal operators, minimal normal extension
Article copyright: © Copyright 1972 American Mathematical Society