x{\to}{\pm}0.$$ Using the integration by parts we deduce the assertion.\
**Proposition 7.4.** *Let conditions $(7.3)$ , $(7.4)$ and $(7.9)$, $(7.10)$ be fulfilled. Then the relation* $$\int_{-\infty}^{\infty}k(t)\mathrm{cos}xtdt=\int_{-\infty}^{\infty}{\nu}^{\prime}(t)
\frac{1-\mathrm{cos}xt}{x^{2}}dt$$ *is true.*\
Relation (7.11) implies that $$\int_{-\infty}^{\infty}k(t)\mathrm{cos}xtdt>0.$$ The kernel $k(x)$ of the operator $S$ admits the representation $$k(x)=\int_{-\infty}^{\infty}m(t)e^{ixt}dt.$$ In view of (7.12) we have $$\mathrm{Re}[m(u)]>0.$$ Due to (7.13) and (7.14) the relation $$(Sf,f)=\int_{-\infty}^{\infty}m(u)|\int_{\Delta}f(t)e^{iut}dt|^{2}du$$ is valid. Hence we have $$-\frac{\pi}{2}{\leq}\mathrm{arg}(Sf,f){\leq}\frac{\pi}{2},
\quad f(t){\in}L^{2}(\Delta).$$ **Proposition 7.5.** *Let conditions $(7.3)$ , $(7.4)$ and $(7.9)$, $(7.10)$ be fulfilled. Then the corresponding operator $B$ is sectorial.*\
*Proof.* Let the function $g(x)$ satisfies conditions (4.1). Then the relation $$(-Lg,g)=(Sg^{\prime},g^{\prime})$$ holds. Equalities (4.3) and (7.17) imply that $$(f,Bf)=(Sg^{\prime},g^{\prime}) ,\quad g=Bf.$$ Inequality (7.1) follows from relations (7.14) and (7.18). Relations (7.16) and (7.18) imply the proposition.\
**Remark 7.1.** The variance damped processes ( Example 2.2.) the normal inverse Gaussian process (Example 2.4.), the Meixner process (Example 2.5.) satisfy the conditions of Proposition 7.5. Hence the corresponding operators $B$ are sectorial.\
2. Now we introduce the notion of the strongly sectorial operators.\
**Definition 7.2.** The sectorial operator $B$ is called *a strongly sectorial* if for some $\beta<1$ relation (7.2) is valid.\
**Proposition 7.6.** *Let the following conditions be fulfilled.\
1). Relations $(7.3)$ , 