Invariant Sübspaces of Certain Linear Operators

This theorem is stronger than a result which Iohvidov [3] derived from the fundamental theorem of [6]. Iohvidov's theorem is so related to Pontrjagin's fundamental theorem that either one can be obtained from the other by a transform analogous to the Cayley transform (see [4]). Pontrjagin's proof of his theorem uses delicate and rather complicated arguments. Kreïn's proof of the theorem stated above is much simpler and consists of an ingenious application of the fixed point principle. In the present note we shall prove two results similar to the Pontrjagin-Iohvidov-Kreïn theorem but of much more general nature, on existence of invariant subspaces of certain linear operators. I t will be seen that the Pontrjagin-Iohvidov-Kreïn theorem can be derived from our Theorem 2. All topological vector spaces considered here are implicitly assumed to be real or complex topological vector spaces satisfying the Hausdorff separation axiom.