A Positive Semi-Definite Square Root Acts on Eigenvectors by the Nonnegative Square Root

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, and let $R$ be a linear operator on $V$ that is self-adjoint and positive semi-definite.

Let $\lambda$ be a real number with $0\le\lambda$, the order being that of the ordered field of real numbers, and let $\mu$ be the unique real number with $0\le\mu$ and $\mu^{2}=\lambda$, which exists and is unique by Existence and Uniqueness of the Nonnegative Square Root of a Nonnegative Real Number; here $\mu^{2}$ abbreviates the product $\mu\cdot\mu$ in the field of real numbers. Real numbers are complex numbers by condition 1 of The Complex Numbers, so $\lambda v$ and $\mu v$ are defined for $v\in V$.

Let $v\in V$ satisfy

$$R(R(v))=\lambda v .$$

Then

$$R(v)=\mu v .$$