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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
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