Linear Algebra Done Right · Problem 7D.9

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

. Suppose every eigenvalue of

T

has absolute value 1 and

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

for every

v \in V

. Prove that

T

is a unitary operator.