Simultaneous completeness in the case of continuous spectrum (Q1084633)

The authors deal with the problem of joint completeness property for contractions on a Hilbert space in the presence of the continuous spectrum. An operator T acting on a Hilbert space H is said to have the joint completeness property if there don't exist non-zero elements of H which are orthogonal to all eigenvectors of T and \(T^*\) corresponding to discrete and absolutely continuous spectra of T and \(T^*\). The study is restricted to completely-non-unitary contractions obtained by one- dimensional perturbations of unitary operators with simple spectra. By using the ''functional model'' and factorising the characteristic function one may define the inner subspaces \((N_ i,N^*_ i)\) and the absolutely continuous subspaces \((N_ e,N^*_ e)\) of the operators T and \(T^*.\) The first result (Theorem 1) is that T has the joint completeness property if the absolutely continuous subspace of T is non-trivial. In the case of a discrete spectrum, a much stronger property holds, namely the ''basisness'' property. The second result (Theorem 2) is that, in the presence of the absolutely continuous spectrum, the intersection \(N_ e\cap N^*_ e\) is non- trivial and consequently the system of the eigenvectors of T and \(T^*\) corresponding to absolutely continuous spectrum is not minimal.