Analytically invariant and bi-invariant subspaces

Herrero, Domingo Antonio; Salinas, Norberto

doi:10.1090/S0002-9947-1972-0312294-9

Analytically invariant and bi-invariant subspaces
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by Domingo Antonio Herrero and Norberto Salinas PDF

Trans. Amer. Math. Soc. 173 (1972), 117-136 Request permission

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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