Given the following statement from a text "Positive operators $A$ have a complete set of eigenvectors, and all their eigenvalues are non-negative. This implies that a positive operator $A$ is Hermitian."
As I understand there are different conventions for defining what a 'positive operator' is. In this context it does not seem sufficient to consider that it is a positive semi-definite, this implies non-negative eigenvalues but not necessarily a complete set of eigenvectors and not necessarily Hermitian. What definition of positive does this statement suggest?
Thanks for any assistance.
Here's a more complete answer than my comment. I believe the notion positivity as defined in your quote can't make sense. Consider the following matrix $$ A = \left( \begin{matrix} 2 & 0 \\ 1 & 1\end{matrix}\right). $$ Note that the two eigenvectors $$\left( \begin{matrix} 0 \\ 1 \end{matrix}\right),\left( \begin{matrix} 1 \\ 1 \end{matrix}\right), $$ have eigenvalues $1$ and $2$. The eigenvectors span the entire space, so it's a matrix with a complete set of eigenvectors and non-negative eigenvalues. Yet it is non-Hermitian. As explained in my comment, being Hermitian is a very desired property for positive matrices to have.