Linear Algebra Done Right · Problem 5E.3

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · commuting-operators

Problem

Suppose S,T∈L(V)S, T \in \mathcal{L}(V) are such that ST=TSST = TS. Suppose p∈P(F)p \in \mathcal{P}(\mathbf{F}). ((a)) Prove that null⁡p(S)\operatorname{null} p(S) is invariant under TT. ((b)) Prove that range⁡p(S)\operatorname{range} p(S) is invariant under TT. See 5.18 for the special case S=TS = T.

Related theory

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