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Invariant subspaces for some operators on locally convex spaces. (English) Zbl 0937.47005
Let $$X$$ be a locally convex Hausdorff space over $$\mathbb C$$ and $$\mathcal L_b(X)$$ the space of all continuous linear operators on X, equipped with the topology of uniform convergence on bounded subsets. The author establishes a total of 9 theorems for the existence of (hyper)invariant subspaces of operators or operator algebras in $$\mathcal L_b(X)$$. Typically, the hypotheses involve some kind of “commutativity with a compact operator”, which makes the assertions reminiscent of the Lomonosov theorem for the Banach space operators. Example: Assume that $$X$$ is complete and $$\mathcal A\subset \mathcal L_b(X)$$ is an operator algebra which is the image of a (possibly nonlinear) continuous operator $$S$$ from a Banach space $$Y$$ into $$\mathcal L_b(X)$$. If there exist linear operators $$K_1,K_2$$ on $$X$$ such that $$K_1$$ is compact, $$K_2$$ is locally bounded and decomposable at $$0$$, and $$\mathcal A K_1\subset K_2\mathcal A$$, then $$\mathcal A$$ has a nontrivial invariant subspace.

##### MSC:
 47A15 Invariant subspaces of linear operators 46A32 Spaces of linear operators; topological tensor products; approximation properties 46A99 Topological linear spaces and related structures
##### Keywords:
invariant subspace; locally convex space; compact operator
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