The resolvent of a generalized boundary value problem for the Sturm- Liouville differential operator equations (Q1122128)

Let H be a separable Hilbert space and A be a strongly positive operator in H. In the space \(L_ 2(0,1;H)\) the operator L is defined by the equalities \({\mathcal D}(L)=W^ 2_ 2(0,1;H(A)\), \(L_{\nu}u=0\), \(\nu =1,2)\), \[ (1)\quad Lu=-u''(x)+Au(x), \] where \[ (2)\quad L_{\nu}u=\gamma_{\nu}u^{(m_{\nu})}(0)+\delta_{\nu}u^{(m_{\nu })}(1)+T_{\nu}u=0, \] \(\nu\) \(=1,2\), \(m_{\nu}=0\), \(T_{\nu}\) is some linear operator on \(W_ q^{m_{\nu}}(0,1;H)\), \(1\leq q<\infty\). It is proved that if \[ \det \left| \begin{matrix} (-1)^{m_ 1}\gamma_ 1 & \delta _ 1 \\ (-1)^{m_ 2}\gamma_ 2 & \delta_ 2 \end{matrix} \right| \neq 0, \] and \(T_{\nu}\) is uniformly continuous in \(\lambda\) then for some \(r>0\) the resolvent set of the operator L contains the set \(\{| \lambda |:| \lambda | >r,| \arg \lambda | >\delta \}\) and the estimate \(\| (\lambda I-L)^{- 1}\| \leq C/| \lambda |\) is true. From this theorem the coercive solvability of the problem (1)-(2) is deduced under some additional conditions the belonging of the resolvent of the operator L to some von Neumann-Schatten class of compact operators is proved.