General scheme of solution interpolation problem in Stieltjes class based on consistent representation of the pair of the non-negative operators. I

Let $G_1$ and $G_2$ — Hilbert space and $K_1\ge0$, $K_2\ge0$ — two non-negative and bounded operators in $G_1$ and $G_2$. Let the operators be connected by the Fundamental Identity $L_2K_2-K_1L_1=v_1u_2^*$. Here $L_2$ and $L_1$ are bounded operators from $G_2$ to $G_1$; and $v_1$ and $u_2$ are bounded operators from Hilbert space $H$ to $G_1$ and $G_2$. We need to find in what conditions operators $K_1$ and $K_2$ posses the consistent integral representation having form
$$ K_r=\int\limits_0^{\infty}R_{T_r}(t)v_rt^{r-t}\,d\sigma(t)v_r^*R^*_{T_r}(t)+W_r+(r-1)FF^*, \quad r=1, 2. $$
Here $T_1=L_2L_1^*$, $T_2=L_1^*L_2$, $v_2=L^*_1v_1$, $W_1\ge0$, $W_1L_1=0$, $L_2F=v_1\gamma^{1/2}$, $W_2\ge0$, $L_2W_2=0$, $R_{T_1}(z)=(I-zT_1)^{-1}$, $R_{T_2}(z)=(I-zT_2)^{-1}$, $\sigma(t)$ — non-decreasing function defined on interval $[0; +\infty)$ and having values in the set of bounded hermitian operators, acting in $H$, $\gamma\geq0$ is operator in space $H$. This problem is shown to have the Stieltjes moment problem in itself, as well as Nevanlinna–Pick and Carathéodory interpolation problems.