we go from ${\cal H}$ to ${\cal H}'$ as prescribed by eq. (\[10\]) but we cannot reverse that equation.

What Eq.(\[10\]) really implies is that the mean energy of $H'$ for the states $\Psi '$ of $\cal H'$ coincides with the mean energy of $H$ for those states of $\cal H$ obtained through the embedding $T$, namely,

$$<\Psi '|H'|\Psi'> =  <T\Psi ' |H| T\Psi'>
\label{11}$$

In other words $ T\Psi'$ is used as a variational state for the eigenstates of the Hamiltonian $H$. In particular $T$ should be chosen in such a way that the states truncated in $\cal H$ , which go down to $\cal H'$, are the ones expected to contribute the most to the ground state of $H$. Thus Eq. (\[10\]) is the basis of the so called variational renormalization group method (VRG). As a matter of fact, the VRG method was the first one to be proposed. The ERG came afterwards as a perturbative extension of the former (see later on).

More generally, any operator $\cal O$ acting in $\cal H$ can be “pushed down" or renormalized to a new operator $\cal O'$ which acts in $\cal H'$ defined by the formula,

$${\cal O}'  = T^{\dag} 