Linear Algebra Done Right · Problem 8A.7

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · range-powers

Problem

Suppose

T \in \mathcal{L}(V)

and

m

is a nonnegative integer such that

\operatorname{range} T^m = \operatorname{range} T^{m+1}.

Prove that

\operatorname{range} T^k = \operatorname{range} T^m

for all

k > m