are called density operators or *states*, since any $\rho$ in $\S(\H)$ determines a normal state $A\mapsto {\mathrm{Tr}}A\rho$ on the algebra $\B(\H)$ . Extreme points of $\S(\H)$ are 1-rank projectors called *pure states*.

Denote by $I_{\H}$ the unit operator on a Hilbert space $\mathcal{H}$ and by ${\mathrm{Id}}_{\mathcal{\H}}$ the identity transformation of the Banach space $\mathfrak{T}(\mathcal{H})$.

We will use the Dirac notations $|\varphi\rangle$, $|\psi\rangle\langle\varphi|$,... for vectors and operators of rank 1 on a Hilbert space (in this notations the action of an operator $|\psi\rangle\langle\varphi|$ on a vector $|\chi\rangle$ gives the vector $\langle\varphi|\chi\rangle\langle\psi|$) [@H-SCI].

We will pay a special attention to the class of unbounded densely defined positive operators on $\H$ having discrete spectrum of finite multiplicity. In Dirac’s notations any such operator ${G}$ can be represented as follows $$\label{H-rep}
G=\sum_{k=0}^{+\infty} E_k|\tau_k\rangle\langle \tau_k|$$ on the domain $\mathcal{D}(G)=\{ \varphi\in\H\,|\,\sum_{k=0}^{+\infty} E^2_k|\langle\tau_k|\varphi\rangle|^2<+\infty\}$, where $\left\{\tau_k\right\}_{k=0}^{+\infty}$ is the orthonormal basis of eigenvectors of ${G}$ corresponding to the nondecreasing sequence $\left\{\smash{E_k}\right\}_{k=0}^{+\infty}$ of eigenvalues tending to $+\infty$. We will use the following (cf.[@W-EBN])

\[D-H\] An operator ${G}$ having representation (\[H-rep\]) is called *discrete*.

The set $\S(\H)$ is compact if and only if $\dim \H<+\infty$. We will use the following

\[Comp\] [@H-c-w-c] *If ${G}$ is a discrete unbounded operator on $\H$ 