Linear Algebra Done Right · Problem 21

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · operators-on-inner-product-spaces

Problem

Suppose

\mathbf{F} = \mathbf{C}

V

is finite-dimensional,

T \in \mathcal{L}(V)

, and all eigenvalues of

T

have absolute value less than

1

. Let

\epsilon > 0

. Prove that there exists a positive integer

m

such that

\|T^m v\| \leq \epsilon \|v\|

for every

v \in V