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If $f$ is an endomorphism of a finite dimensional vector space over a field$K$ then an invariant subspace $X \subseteq V$ 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. Inthis paper we obtain a description of the set of all characteristicnon-hyperinvariant subspaces for nilpotent maps $f$ with exactly two unrepeatedelementary divisors.

Linear transformations with characteristic subspaces that are not hyperinvariant