Linear Algebra Done Right · Problem 7A.27

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Suppose

T \in \mathcal{L}(V)

is normal. Prove that

\operatorname{null} T^k = \operatorname{null} T \quad \text{and} \quad \operatorname{range} T^k = \operatorname{range} T

for every positive integer

k