Linear Algebra Done Right · Problem 5E.6

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · commuting-operators

Problem

Suppose that

V

is a nonzero finite-dimensional complex vector space and

S, T \in \mathcal{L}(V)

commute. Prove that there exist

\alpha, \lambda \in \mathbf{C}

such that

\operatorname{range}(S - \alpha I) + \operatorname{range}(T - \lambda I) \neq V.