Operators Diagonal in an Orthonormal Basis

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space. Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $e\in V^{n}$ be an $n$-tuple in $V$ that is an orthonormal basis of $V$, with components $e_{k}$. Let $\mu\in\mathbb{C}^{n}$ be an $n$-tuple with components $\mu_{k}$ in the field $\mathbb{C}$ of complex numbers.

Sums of vectors are finite sums in $V$. Let $R$ be the map from $V$ to $V$ given by

$$R(x)=\sum_{k=1}^{n}\bigl(\mu_{k}\langle e_{k},x\rangle\bigr)e_{k}\qquad\text{for }x\in V.$$

Then the following hold.

1. (Linearity and action on the basis) $R$ is a linear operator on $V$, and $R(e_{j})=\mu_{j}e_{j}$ for every $j\in[n]$.

2. (Self-adjointness) If $\mu_{k}$ is a real number for every $k\in[n]$, then $R$ is self-adjoint.