A theorem on uniform correctness of a Cauchy problem and its application (Q1067586)

The author considers an operator \(A=i(d/dt)\) in the complex space \(L_ 2(0,1)\), with the domain \(D_ A=Ker {\hat \phi}\) for \({\hat \phi}\in L[W^{1,2}(0,1),R]\setminus L[L_ 2(0,1),R]\). The necessary and sufficient conditions are given under which the operator -iA generates a uniformly correct Cauchy problem. As an example of an application he shows that tf is not possible to invert the following Krein theorem about extension of a generator of \(C_ 0\)-semigroup: ''If B is an infinitesimal generator of a uniformly correct Cauchy problem of the type \(\omega\) in Hilbert space H, than the operator \(B-\omega_ 1I\) admits a maximal dissipative extension \(B-\omega_ 1I\) \((\omega_ 1>\omega)\) which acts in the space \(H_ 1\supsetneqq H\), with \(D_ B\subset H.''\) Namely, the author constructs an example of a closed operator which satisfies the conclusions of the Krein theorem but which is not an infinitesimal generator of any correct Cauchy problem. Cf. \textit{S. G. Krein}, Linear differential equations in Banach space, (Russian) (1967; Zbl 0172.419).