on $\HH$ such that $$\label{defining property of $1$-pdm}
    \langle g    | \gamma_\Delta f   \rangle \equals \Tr\big(\Delta \hat{a}^*(f)\hat{a}(g) \big)$$ for all $f,g \in \HH$. The preceding formula implies that $\gamma_{\Delta}$ is a positive-semidefinite contraction. Its eigenvectors are called “natural orbitals" of $\Delta$, and the corresponding eigenvalues are the “natural occupation numbers" of $\Delta$.

If $h \in \HH$ is a unit vector, the diagonal matrix element $ \langle h | \gamma_\Delta h  \rangle$ of $\gamma_\Delta $ is the probability that the orbital $h$ is occupied when the system is in the state represented by $\Delta$. This is because the operator $\hat{a}^*(h)\hat{a}(h)$ corresponds to the physical observable of orbital $h$’s occupation, which takes the values $0$ and $1$. The expected value of this observable, the probability that orbital $h$ is occupied, is therefore $\Tr\big( \Delta \hat{a}(h)^*\hat{a}(h) \big)$, and this equals $\langle h|\gamma_{\Delta} h \rangle $ by definition.

We are especially interested in the class of density operators on $\FFF = \FFF(\HH)$ that represent states of finite average particle number. We shall denote this class by $\DD(\FFF)$. If $\Delta$ is a density operator on $\FFF$, then the average number of particles in the state represented by $\Delta$ equals the trace of its $1$-pdm. Thus, 