Linear Algebra Done Right · Problem 17

Inner Product Spaces — Sheldon Axler

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

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

V

is finite-dimensional. Prove that if

T

is an operator on

V

such that

1

is the only eigenvalue of

T

and

\|Tv\| \leq \|v\|

for all

v \in V

, then

T

is the identity operator.