$f_{1}$ that $$Bf_{1}=\lambda_{1}f_{1}+g_{1}.$$ In this case the relations $$(Bf_{1},h_{1})=\lambda_{1}(f_{1},h_{1})+(g_{1},h_{1})=\lambda_{1}(f_{1},h_{1})$$are true. Hence $(g_{1},h_{1})=0.$ The last relation contradicts condition (9.2). It proves equality (9.13).\
Relation (8.9) implies that $$\Phi(0,x){\in}L^{p}(\Delta).$$ We denote by $W(B)$ the numerical range of $B$. The closure of the convex hull of $W(B)$ is situated in the sector (9.7). Hence the estimation $$||(B^{\star}-zI)^{-1}||_{p}{\leq}M/|z|,\quad z{\in}\Gamma_{\epsilon}$$ is true (see \[26\] for the Hilbert case $p=2$ and \[16\],\[28\] for the Banach space $p{\geq}1$). By $||B||_{p}$ we denote the norm of the operator $B$ in the space $L^{p}(\Delta)$.\
It follows from estimation(9.17) that the integral J exists.\
Among the numbers $\lambda_{k}$ we choose for which $\mathrm{Re}(1/{\lambda}_{k}),\quad (1{\leq}k{\leq}m)$ has the smallest value $\delta$. Among the obtained numbers we choose $\mu_{k},\quad (1{\leq}k{\leq}\ell)$ the indexes $n_{k}$ of which have the largest value $n$. We deduce from (9.10)-(9.12) that $$p(t,\Delta)=e^{-t{\delta}}t^{n}
 [\sum_{k=1}^{\ell}e^{-t/{\mu}_{k}}c_{k}+o(1)],\quad t{\to}\infty.$$ We note that the function $$Q(t)= \sum_{k=1}^{\ell}e^{it\mathrm{Im}(\mu_{k}^{-1})}c_{k}$$ is almost periodic (see \[14\]). Hence in view of (9.18) and the inequality\
$p(t,\Delta)>0,\quad t{\geq}0$ the following relation $$Q(t){\geq}0,\quad -\infty<t<\infty$$ is valid.\
First we assume that at least one of the inequalities $$\delta<{\lambda}_{1}^{-1},\quad n>1$$ is true. Using (9.21) and the inequality $$\lambda_{1}{\geq}\lambda_{k},\quad k=2,3,...$$we have $$\mathrm{Im}{\mu}_{j}^{-1}{\ne}0,\quad 1{\leq}j{\leq}\ell.$$ It follows from (9.19) that $$c_{j}=\mathrm{lim}\frac{1}{2T}\int_{-T}^{T}Q(t)e^{-it(\mathrm{Im}{\mu}_{j}^{-1})}dt,\quad
T{\to}\infty.$$In view of (9.20) we obtain 