An operator without invariant subspaces on a nuclear Fréchet space. (English) Zbl 0553.47002

Atzmon, Aharon

Ann. Math. (2) 117, 669-694 (1983).

The author constructed an infinite dimensional nuclear Fréchet space F (which is also a nuclear countably Hilbert space, in the sense of Gelfand and Vilenkin), a backward shift T on f, \((Tc)_ j=c_{j+1}\), \(j=0,1,..\). and proved that T has no invariant subspaces, but \(T^ 2\) has an invariant subspace (theorem 2.9). The method he used to obtain the assertion is the following. He showed that the dual space of F can be identified with a certain topological algebra E, of analytic functions with varying domains and Theorem 2.9 is equivalent to that E has no closed nontrivial ideals (Theorem 3.7), then he proved Theorem 3.7.

Reviewer: Lu Shijie

MSC:

47A15	Invariant subspaces of linear operators
46A13	Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11	Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46H10	Ideals and subalgebras
46A04	Locally convex Fréchet spaces and (DF)-spaces

Keywords:

nuclear Fréchet space; nuclear countably Hilbert space; backward shift; invariant subspace; dual space; closed nontrivial ideals