there is one and only one function $$\psi(x,s)=(I+sB^{\star})^{-1}\Phi(0,x) , \quad 0{\leq}s<s_{0},$$ which satisfies relation $(5.22)$.*\
*Proof.* In view of (4.4) we have $$-BL_{\Delta}f=f,\quad f{\in}C_{\Delta}.$$Relations (5.23) and (5.24) imply that $$((sI-L_{\Delta})f,\psi(x,s))_{\Delta}=-((I+sB)L_{\Delta}f,\psi)_{\Delta}=-(L_{\Delta}f,\Phi(0,x))_{\Delta}.$$ Since $\Phi(0,x)=B^{\star}\delta(x),$ ($\delta(x)$ is the Dirac function) then according to (5.23) and (5.25) relation (5.22) is true.\
Let us suppose that in $L(\Delta)$ there is another function $\psi_{1}(x,s)$ satisfying (5.22). Then the equality $$((sI-L_{\Delta})f,\phi(x,s))_{\Delta}=0,\quad
\phi=\psi-\psi_{1}$$ is valid. We write relation (5.26) in the form $$(L_{\Delta}f,(I+sB^{\star})\phi)_{\Delta}=0.$$ Due to (4.4) the range of $L_{\Delta}$ is dense in $L^{p}(\Delta).$ Hence in view of (5.27) we have $\phi=0.$ The theorem is proved.\
The analytical apparatus for the construction and investigation of the function $\psi(x,s)$ is based on relation (5.22) and properties of the quasi-potential $B$. In the following three sections we shall investigate the properties of the operator $B$.

Non-negativity of the kernel $\Phi(x,y)$
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In this section we deduce the following important property of the kernel $\Phi(x,y)$.\
**Proposition 6.1.** *Let the density $\rho(x,t)$ of Levy process $X_{t}$ be continuous $(t>0)$ and let the corresponding quasi-potential $B$ satisfy conditions $(4.6)-(4.8)$ of Definition $4.3.$ Then the kernel $\Phi(x,y)$ is non-negative i.e.* $$\Phi(x,y){\geq}0.$$ *Proof.* In view of (5.9) and (5.10) we have $ \psi(x,s,u){\geq}0$. Relation (5.19)implies 