A criterion for nuclearity of the resolvent of a certain differential operator (Q760568)

Let \(I_+=[0,\infty)\) and p(x), q(x) and r(x) be nonnegative twice continuously differentiable functions on \(I_+\), p(x), r(x) are positive for \(x>0\). The authors consider the Friedrichs expansions L in \(L_ 2(I_+)\) of the operator \(L_ 0\), defined on \(C^ 0_ 2(I_+)\) by \[ (1)\quad L_ 0u=-p(x)[r^ 2(x)(p(x)u)']'+q^ 2(x)u. \] Using some of \textit{M. Otelbaev}'s results [ibid. 25, 569-572 (1979; Zbl 0409.47028)], the authors prove an inequality and as a consequence, they obtain the fact that if \(L^{-1}\) is completely continuous, then \(L^{-1}\) is a nuclear operator if it fulfils further conditions. Some other results are also established.