Operators on Complex Vector Spaces — Sheldon Axler
Mathematics — Linear Algebra · nilpotent-operators
Problem
(a) Give an example of nilpotent operators S,T on the same vector space such that neither S+T nor ST is nilpotent.
(b) Suppose S,T∈L(V) are nilpotent and ST=TS. Prove that S+T and ST are nilpotent.