On the Hilbert space $\mathbb{R}^2$, what is a concrete example of a positive operator-valued measure that is not a projection-valued measure?

Question

I am looking for a purely mathematical example. I tried looking for a set of symmetric matrices $\{F_1,F_2\}$ such that $F_1+F_2=I$ but I cannot seem to find an example.

Answer 1

To do an example as you want, you take $F_1$ to be any positive matrix with $F_1\leq I$ (equivalently, $F_1$ is selfadjoint, and its eigenvalues are in $[0,1]$), and then take $F_2=I-F_1$. As a simplest example you could take $$ F_1=\begin{bmatrix}1/2&0\\0&1/3\end{bmatrix},\ \ \ F_2=\begin{bmatrix}1/2&0\\0&2/3\end{bmatrix}. $$