Linear Algebra Done Right · Problem 7C.16

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

T

is a positive operator on

V

. Prove that

\operatorname{null} \sqrt{T} = \operatorname{null} T \quad \text{and} \quad \operatorname{range} \sqrt{T} = \operatorname{range} T.