Lomonosov’s Theorem on Invariant Subspaces

The Invariant Subspace Problem asks whether or not a bounded linear operator A on a Banach space X must admit a nontrivial, proper invariant subspace.

In 1954, Aronszajn and Smith proved that the answer is “Yes” if A is compact. In 1966, Bernstein and Abraham Robinson proved, using techniques from non-standard analysis,

that the answer is “Yes” if some polynomial in A is compact. In both cases the proofs were quite involved so it came as quite a shock to all concerned when, in 1973, Lomonosov

obtained a much more general theorem whose proof was, by comparison, almost trivial. I will prove Lomonosov’s theorem from scratch.

Incidentally, in 1976, Per Enflo showed that answer is “No” in general.