Linear Algebra Done Right · Problem 7C.22

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

T \in \mathcal{L}(V)

is a positive operator and

u \in V

is such that

\|u\| = 1

and

\|Tu\| \geq \|Tv\|

for all

v \in V

with

\|v\| = 1

. Show that

u

is an eigenvector of

T

corresponding to the largest eigenvalue of

T