Linear Algebra Done Right · Problem 7A.19

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Suppose

T \in \mathcal{L}(V)

and

\|T^*v\| \leq \|Tv\|

for every

v \in V

. Prove that

T

is normal. This exercise fails on infinite-dimensional inner product spaces, leading to what are called hyponormal operators, which have a well-developed theory.

Related theory

normal-operators

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