Linear transformations with characteristic subspaces that are not hyperinvariant

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If f is an endomorphism of a finite dimensional vector space over a field K then an invariant subspace X V is called hyperinvariant (respectively, characteristic) if X is invariant under all endomorphisms (respectively, automorphisms) that commute with f. According to Shoda (Math. Zeit. 31, 611--624, 1930) only if |K| = 2 then there exist endomorphisms f with invariant subspaces that are characteristic but not hyperinvariant. In this paper we obtain a description of the set of all characteristic non-hyperinvariant subspaces for nilpotent maps f with exactly two unrepeated elementary divisors.